Siegel automorphic form corrections of some lorentzian Kac-Moody Lie algebras
Gritsenko, V A; Gritsenko, Valeri A; Nikulin, Viacheslav V
1995-01-01
We find automorphic form corrections (which are the generalized Lorentzian Kac--Moody Lie superalgebras) for two elliptic Lorentzian Kac--Moody Lie algebras of the rank 3 with a lattice Weyl vector, and calculate multiplicities of their simple and arbitrary imaginary roots. These Kac--Moody Lie algebras are defined by hyperbolic (i.e. with exactly one negative square) symmetric generalized Cartan matrices G_1=\\pmatrix 2&-2&-2\\\\-2&2&-2\\\\-2&-2&2\\endpmatrix, G_2=\\pmatrix 4&-4&-12&-4\\\\-4&4&-4&-12\\\\ -12&-4&4&-4\\\\-4&-12&-4&4\\endpmat rix of the rank 3. Both these algebras have elliptic type (i.e. their Weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces) and have a lattice Weyl vector. The correcting automoprhic forms are Siegel modular forms: the classical Siegel cusp form of weight 5 (with a multiplier system) which is the product of ten even theta-constants (for the algebra G_1) and a cusp form of weight...
Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras
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Dmitry Fuchs
2008-08-01
Full Text Available We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of sl_2 (Theorem 3. The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986, no. 2, 103-113]. In the simpler case of A_1^1 the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989, no. 2, 154-156].
Institute of Scientific and Technical Information of China (English)
刘斌
2001-01-01
给出了超双曲型(以下简称SH-型)Kac-Moody Lie代数的结构定理及部分Dykin结构图，同时也证明了所有SH-型广义Cartan矩阵A=(aij)l×l是非奇异的，且有惯性指数(1-1，1)。%The structure theory of the super hyperbolic type Kac - Moody Lie algebra and its part Dykin groph are given. It is proved that this kind of cartan matrices A= (aij)l× lis nonsingular and its inertial index is (1 - 1,1).
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
Zheng-xin CHEN; Ya-nan LIN
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)C1/I(A) of complex degenerate composition Lie algebras L(A)C1 by some ideals, where L(A)C1 is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)C1/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)C1 generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)C1 generated by simple A-modules.
Chromatic polynomials of graphs from Kac-Moody algebras
Venkatesh, R.; Viswanath, Sankaran
2013-01-01
We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie algebra evaluated on the sum of the simple roots. Applying the Peterson recurrence formula for root multiplicities, we obtain a new realization of the chromatic polynomial as a weighted sum of paths in the bond lattice of G.
Unique factorization of tensor products for Kac-Moody algebras
Venkatesh, R.; Viswanath, Sankaran
2012-01-01
We consider integrable, category O-modules of indecomposable symmetrizable Kac-Moody algebras. We prove that unique factorization of tensor products of irreducible modules holds in this category, upto twisting by one dimensional modules. This generalizes a fundamental theorem of Rajan for finite dimensional simple Lie algebras over C. Our proof is new even for the finite dimensional case, and uses an interplay of representation theory and combinatorics to analyze the Kac-Weyl character formula.
Automorphic Forms and Lorentzian Kac-Moody Algebras, 1
Gritsenko, V A; Gritsenko, Valeri A.; Nikulin, Viacheslav V.
1996-01-01
Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac--Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac--Moody algebras, and we formulate finiteness Conjecture for ...
Kac-Moody Algebras and Controlled Chaos
Wesley, D H
2007-01-01
Compactification can control chaotic Mixmaster behavior in gravitational systems with p-form matter: we consider this in light of the connection between supergravity models and Kac-Moody algebras. We show that different compactifications define "mutations" of the algebras associated with the noncompact theories. We list the algebras obtained in this way, and find novel examples of wall systems determined by hyperbolic (but not strictly hyperbolic) algebras. Cosmological models with a smooth pre-big bang phase require that chaos is absent: we show that compactification alone cannot eliminate chaos in the simplest compactifications of the heterotic string on a Calabi-Yau, or M theory on a manifold of G_2 holonomy.
Generalized Kac-Moody Algebras from CHL dyons
Govindarajan, Suresh
2009-01-01
We show the existence of a family of generalized Kac-Moody(GKM) superalgebras, G_N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Delta_{k/2}(Z), for (N,k)=(1,10),(2,6),(3,4),(5,2). The square of the automorphic form is related to the generating function of the degeneracy of CHL dyons in asymmetric Z_N-orbifolds of the heterotic string compactified on T^6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Delta_{k/2}(Z), seem to arise as additive lifts of weak Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, G_1, leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the `algebra of BPS states' in CHL compacti...
Generalized Kac-Moody algebras from CHL dyons
Govindarajan, Suresh; Gopala Krishna, K.
2009-04-01
We provide evidence for the existence of a family of generalized Kac-Moody (GKM) superalgebras, Script GN, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Δk/2(Z), for (N, k) = (1, 10), (2, 6), (3, 4), and possibly (5, 2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric Bbb ZN-orbifolds of the heterotic string compactified on T6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Δk/2(Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, Script G1, leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the `algebra of BPS states' in CHL compactifications.
The Kostant partition functions for twisted Kac-Moody algebras
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Ranabir Chakrabarti
2000-01-01
Full Text Available Employing the method of generating functions and making use of some infinite product identities like Euler, Jacobi's triple product and pentagon identities we derive recursion relations for Kostant's partition functions for the twisted Kac-Moody algebras.
The Kostant partition functions for twisted Kac-Moody algebras
Ranabir Chakrabarti; Santhanam, Thalanayar S.
2000-01-01
Employing the method of generating functions and making use of some infinite product identities like Euler, Jacobi's triple product and pentagon identities we derive recursion relations for Kostant's partition functions for the twisted Kac-Moody algebras.
Generalized conformal realizations of Kac-Moody algebras
Palmkvist, Jakob
2009-01-01
We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n =1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3×3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n =2. Moreover, we obtain their infinite-dimensional extensions for n ≥3. In the case of 2×2 matrices, the resulting Lie algebras are of the form so(p +n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).
Automorphic Forms and Lorentzian Kac-Moody Algebras, 2
Gritsenko, V A; Gritsenko, Valeri A.; Nikulin, Viacheslav V.
1996-01-01
We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). The data for these liftings are Jacobi forms of integral and half-integral indices. In particular, we get modular forms which are generalizations of the Dedekind eta-function. Some of these forms define automorphic corrections of Lorentzian Kac-Moody algebras with hyperbolic generalized Cartan matrices of rank three classified in Part I of this paper. We also construct many automorphic forms which give discriminants of moduli of K3 surfaces with conditions on Picard lattice. These results are important for Mirror Symmetry and theory of Lorentzian Kac-Moody algebras.
Representations of general linear groups and categorical actions of Kac-Moody algebras
Losev, Ivan
2012-01-01
This is an expanded version of the lectures given by the author on the 3rd school "Lie algebras, algebraic groups and invariant theory" in Togliatti, Russia. In these notes we explain the concept of a categorical Kac-Moody action by studying an example of the category of rational representations of a general linear group in positive characteristic. We also deal with some more advanced topics: a categorical action on the polynomial representations and crystals of categorical actions.
K3 surfaces, lorentzian Kac-Moody algebras and mirror symmetry
Gritsenko, V A; Gritsenko, Valeri A; Nikulin, Viacheslav V
1995-01-01
We consider the variant of Mirror Symmetry Conjecture for K3 surfaces which relates "geometry" of curves of a general member of a family of K3 with "algebraic functions" on the moduli of the mirror family. Lorentzian Kac--Moody algebras are involved in this construction. We give several examples when this conjecture is valid.
Affine Kac-Moody algebras, CHL strings and the classification of tops
Bouchard, Vincent; Bouchard, Vincent; Skarke, Harald
2003-01-01
Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group.
Variable coefficient Davey-Stewartson system with a Kac-Moody-Virasoro symmetry algebra
Güngör, F.; Özemir, C.
2016-06-01
We study the symmetry group properties of the variable coefficient Davey-Stewartson (vcDS) system. The Lie point symmetry algebra with a Kac-Moody-Virasoro (KMV) structure is shown to be isomorphic to that of the usual (constant coefficient) DS system if and only if the coefficients satisfy some conditions. These conditions turn out to coincide with those for the vcDS system to be transformable to the DS system by a point transformation. The equivalence group of the vcDS system is applied to pick out the integrable subsystems from a class of non-integrable ones. Additionally, the full symmetry group of the DS system is derived explicitly without exponentiating its symmetry algebra. Lump solutions (rationally localized in all directions in ℝ2) introduced by Ozawa for the DS system are shown to hold even for the vcDS system precisely when the system belongs to the integrable class, i.e., equivalent to the DS system. These solutions can be used for establishing exact blow-up solutions in finite time in the space L2(ℝ2) in the focusing case.
Geometric Kac-Moody Modularity
Lynker, M; Lynker, Monika; Schimmrigk, Rolf
2004-01-01
It is shown how the arithmetic structure of algebraic curves encoded in the Hasse-Weil L-function can be related to affine Kac-Moody algebras. This result is useful in relating the arithmetic geometry of Calabi-Yau varieties to the underlying exactly solvable theory. In the case of the genus three Fermat curve we identify the Hasse-Weil L-function with the Mellin transform of the twist of a number theoretic modular form derived from the string function of a non-twisted affine Lie algebra. The twist character is associated to the number field of quantum dimensions of the conformal field theory.
The Igusa modular forms and "the simplest" lorentzian Kac-Moody algebras
Gritsenko, V A; Gritsenko, Valeri A; Nikulin, Viacheslav V
1996-01-01
We find automorphic corrections for the Lorentzian Kac--Moody algebras with the simplest generalized Cartan matrices of rank 3: 2 0 -1 2 -2 -1 A_{1,0} = 0 2 -2 and A_{1,I} = -2 2 -1 -1 -2 2 -1 -1 2 For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form \\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30 with respect to a 2-congruence subgroup. We find infinite product or sum expansions for these forms. Our method of construction of \\chi_{35} leads to the direct construction of Siegel modular forms by infinite product expansions, whose divisors are the Humbert surfaces with fixed discriminants. Existence of these forms was proved by van der Geer in 1982 using some geometrical consideration. We announce a list of all hyperbolic symmetric generalized Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a lattice Weyl vector, and A contains the affine submatrix \\tilde{A}_1.
The Lie algebra of the N=2-string
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Kugel, K.
2006-07-01
The theory of generalized Kac-Moody algebras is a generalization of the theory of finite dimensional simple Lie algebras. The physical states of some compactified strings give realizations of generalized Kac-Moody algebras. For example the physical states of a bosonic string moving on a 26 dimensional torus form a generalized Kac-Moody algebra and the physical states of a N=1 string moving on a 10 dimensional torus form a generalized Kac-Moody superalgebra. A natural question is whether the physical states of the compactified N=2-string also realize such an algebra. In this thesis we construct the Lie algebra of the compactified N=2-string, study its properties and show that it is not a generalized Kac-Moody algebra. The Fock space of a N=2-string moving on a 4 dimensional torus can be described by a vertex algebra constructed from a rational lattice of signature (8,4). Here 6 coordinates with signature (4,2) come from the matter part and 6 coordinates with signature (4,2) come from the ghost part. The physical states are represented by the cohomology of the BRST-operator. The vertex algebra induces a product on the vector space of physical states that defines the structure of a Lie algebra on this space. This Lie algebra shares many properties with generalized Kac-Moody algebra but we will show that it is not a generalized Kac-Moody algebra. (orig.)
Directory of Open Access Journals (Sweden)
Daisuke Yamakawa
2010-10-01
Full Text Available To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlevé equations is slightly different from (but equivalent to Okamoto's.
Structure of Solvable Quadratic Lie Algebras
Institute of Scientific and Technical Information of China (English)
ZHU Lin-sheng
2005-01-01
@@ Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics[10,12,13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras.
Kac-Moody exceptional E12 from simplictic tiling
Energy Technology Data Exchange (ETDEWEB)
El Naschie, M.S. [Frankfurt Institute for the Advancement of Fundamental Scientific Research, University of Frankfurt (Germany); Department of Physics, University of Alexandria (Egypt)], E-mail: chaossf@aol.com
2009-08-30
We give various derivations for the order of a new non classical exceptional Lie group E12. We start from the simplest polyhedra of ordinary three dimensional space and arrive at the exact integer value vertical bar E12 vertical bar = 685. Subsequently we show that a corresponding infinite dimensional but hierarchal KAC-Moody algebra called 4D fusion algebra leads to an exact transfinite dimension equal to Dim E12=(5)({alpha}-bar{sub o})=685.410197, where {alpha}-bar{sub o}=137.082039325 is the E-infinity electromagnetic fine structure constant.
Kac-Moody Symmetry in Hosotani Model
Shiraishi, Kiyoshi
2012-01-01
The symmetry of the massive tower of fields in higher-dimensional Yang-Mills theory compactified on a space-time of the form M_d x S^1 is clarified. The transformations form a loop algebra, a class of Kac-Moody algebras. Since the symmetry is spontaneously broken, vector fields "eat" Goldstone bosons and acquire masses. The field of zero-mass mode can also become massive provided that the field of the internal component develops a vacuum expectation value. The relation between the "restoration" of the symmetry in massive modes and the gauge transformation of the zero-mode vacuum field is discussed.
Homogeneous Construction of the Toroidal Lie Algebra of Type A1
Institute of Scientific and Technical Information of China (English)
Haifeng Lian; Cui Chen; Qinzhu Wen
2007-01-01
In this paper,we consider an analogue of the level two homogeneous construc-tion of the affine Kac-Moody algebra A1(1) by vertex operators.We construct modules for the toroidal Lie algebra and the extended toroidal Lie algebra of type A1.We also prove that the module is completely reducible for the extended toroidal Lie algebra.
Lie Point Symmetries of Differential-Difference Equations
Institute of Scientific and Technical Information of China (English)
DING Wei; TANG Xiao-Yan
2004-01-01
In this paper, the classical Lie group approach is extended to find some Lie point symmetries of differentialdifference equations. It reveals that the obtained Lie point symmetries can constitute a Kac-Moody-Virasoro algebra.
Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories
Bao, Ling
2013-01-01
Kac-Moody groups $G$ over $\\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of $G(\\mathbb{Z})$, due to Tits, are functorial and not amenable to computation or applications. We construct Kac-Moody groups over $\\mathbb{R}$ and $\\mathbb{Z}$ using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We extend a construction of Eisenstein series on finite dimensional semisimple algebraic groups using representation theory, which appeared in the context of superstring theory, to general Kac-Moody groups. This coincides with a generalization of Garland's Eisenstein series on affine Kac-Moody groups to general Kac-Moody groups and includes Eisenstein series on $E_{10}$ and $E_{11}$. For finite dimensional groups, Eisenstein series encode the quantum corrections in string theory and supergravity theories. T...
Extremal projectors for contragredient Lie (super)symmetries (short review)
Tolstoy, V N
2010-01-01
A brief review of the extremal projectors for contragredient Lie (super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie superalgebras, infinite-dimensional affine Kac-Moody algebras and superalgebras, as well as their quantum $q$-analogs) is given. Some bibliographic comments on the applications of extremal projectors are presented.
Permutation Weights and Modular Poincare Polynomials for Affine Lie Algebras
Gungormez, M
2010-01-01
Poincare Polynomial of a Kac-Moody Lie algebra can be obtained by classifying the Weyl orbit $W(\\rho)$ of its Weyl vector $\\rho$. A remarkable fact for Affine Lie algebras is that the number of elements of $W(\\rho)$ is finite at each and every depth level though totally it has infinite number of elements. This allows us to look at $W(\\rho)$ as a manifold graded by depths of its elements and hence a new kind of Poincare Polynomial is defined. We give these polynomials for all Affine Kac-Moody Lie algebras, non-twisted or twisted. The remarkable fact is however that, on the contrary to the ones which are classically defined,these new kind of Poincare polynomials have modular properties, namely they all are expressed in the form of eta-quotients. When one recalls Weyl-Kac character formula for irreducible characters, it is natural to think that this modularity properties could be directly related with Kac-Peterson theorem which says affine characters have modular properties. Another point to emphasize is the rel...
Jacobson, Nathan
1979-01-01
Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its
Masoero, Davide; Valeri, Daniele
2015-01-01
We assess the ODE/IM correspondence for the quantum $\\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\\mathfrak{g}}^{(1)}$, and constructing the relevant $\\Psi$-system among subdominant solutions. We then use the $\\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
Masoero, Davide; Raimondo, Andrea; Valeri, Daniele
2016-09-01
We assess the ODE/IM correspondence for the quantum g -KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra g^{(1)} , and constructing the relevant {Ψ} -system among subdominant solutions. We then use the {Ψ} -system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum g -KdV model. We also consider generalized Airy functions for twisted Kac-Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
Laplace Operators of Infinite-Dimensional Lie Algebras and Theta Functions
Kac, Victor G.
1984-01-01
Until recently, the generalized Casimir operator constructed by Kac [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70] has been the only known element of the center of a completion of the enveloping algebra of a Kac-Moody algebra. It has been conjectured [Deodhar, V. V., Gabber, O. & Kac, V. G. (1982) Adv. Math. 45, 92-116], however, that the image of the Harish-Chandra homomorphism contains all theta functions defined on the interior of the complexified Tits cone and hence separates the orbits of the Weyl group. Developing the ideas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1983) Dokl. Akad. Nauk SSSR 269, 1057-1060], I prove this conjecture. Another application of this method is the Chevalley type restriction theorem for simple finite-dimensional Lie superalgebras.
Solvable quadratic Lie algebras
Institute of Scientific and Technical Information of China (English)
ZHU; Linsheng
2006-01-01
A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.
Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
Fleig, Philipp; Persson, Daniel
2013-01-01
Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on E_9(R), E_10(R) and E_11(R) corresponding to certain degenerate principal series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings R^4 and \\partial^{4} R^4 coming from 1/2-BPS and 1/4-BPS instantons, respectively. This suggests that there exist minimal and next-to-minimal unipotent automorphic representations of the associated Kac-Moody groups to which these special Eisenstein series are attached. We also provide complete explicit expressions for degenerate Whittaker vectors of minimal Eisenstein series on E_6(R), E_7(R) and E_8(R) that have not appeared in the literature before.
Quantum Lie theory a multilinear approach
Kharchenko, Vladislav
2015-01-01
This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely accepted. In this book, an alternative approach is developed that includes multivariable operations. Among the problems discussed are the following: a PBW-type theorem; quantum deformations of Kac--Moody algebras; generic and symmetric quantum Lie operations; the Nichols algebras; the Gurevich--Manin Lie algebras; and Shestakov--Umirbaev operations for the Lie theory of nonassociative products. Opening with an introduction for beginners and continuing as a textbook for graduate students in physics and mathematics, the book can also be used as a reference by more advanced readers. With the exception of the introductory chapter, the content of this monograph has not previously appeared in book form.
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Palmkvist, Jakob, E-mail: palmkvist@ihes.fr [Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, FR-91440 Bures-sur-Yvette (France)
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices
Institute of Scientific and Technical Information of China (English)
Li Xia YE; Zhi Xiang WU; Xue Feng MEI
2007-01-01
Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. Inthis paper, we define a d-type weak quantum generalized Kac-Moody algebra wUdq(y), which is a weakHopf algebra. We also study the highest weight module over the weak quantum algebra wUdq(y) andWeak A-forms of wUdq(y).
Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups
Dechant, Pierre-Philippe; Twarock, Reidun
2011-01-01
Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H_2, H_3 and H_4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H_3 generate (twist) translations along 2-, 3- and 5-fold axes of icosahedral symmetry and classify these translations in terms of Fibonacci recursion relations, thus providing a framework to explain results of Keef et al and Wardman at the group level. Finally, we extend this classification to the case of the non-crystallographic Coxeter groups H_2 and H_4. These results should have applications in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).
Lie groups and Lie algebras for physicists
Das, Ashok
2015-01-01
The book is intended for graduate students of theoretical physics (with a background in quantum mechanics) as well as researchers interested in applications of Lie group theory and Lie algebras in physics. The emphasis is on the inter-relations of representation theories of Lie groups and the corresponding Lie algebras.
Introduction to quantum Lie algebras
Delius, G W
1996-01-01
Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in h. They are derived from the quantized enveloping algebras \\uqg. The quantum Lie bracket satisfies a generalization of antisymmetry. Representations of quantum Lie algebras are defined in terms of a generalized commutator. In this paper the recent general results about quantum Lie algebras are introduced with the help of the explicit example of (sl_2)_h.
Laplace operators of infinite-dimensional Lie algebras and theta functions
Kac, Victor G.
1984-01-01
Until recently, the generalized Casimir operator constructed by Kac [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70] has been the only known element of the center of a completion of the enveloping algebra of a Kac-Moody algebra. It has been conjectured [Deodhar, V. V., Gabber, O. & Kac, V. G. (1982) Adv. Math. 45, 92-116], however, that the image of the Harish-Chandra homomorphism contains all theta functions defined on the interior of the complexified Tits cone and hence separates the orbits ...
Lie algebraic noncommutative gravity
Banerjee, Rabin; Mukherjee, Pradip; Samanta, Saurav
2007-06-01
We exploit the Seiberg-Witten map technique to formulate the theory of gravity defined on a Lie algebraic noncommutative space-time. Detailed expressions of the Seiberg-Witten maps for the gauge parameters, gauge potentials, and the field strengths have been worked out. Our results demonstrate that notwithstanding the introduction of more general noncommutative structure there is no first order correction, exactly as happens for a canonical (i.e. constant) noncommutativity.
Weak Lie symmetry and extended Lie algebra
Energy Technology Data Exchange (ETDEWEB)
Goenner, Hubert [Institute for Theoretical Physics, Friedrich-Hund-Platz 1, University of Goettingen, D-37077 Gottingen (Germany)
2013-04-15
The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).
Two-point spin-1/2-spin-1/2 sl(2,bfC) conformal Kac-Moody blocks on the torus and their monodromies
Energy Technology Data Exchange (ETDEWEB)
Smyrnakis, J.M. [Columbia Univ., New York, NY (United States). Dept. of Mathematics
1995-10-02
Two issues of the SU(2) Wess-Zumino-Witten model are examined here, namely the computation of the untwisted conformal Kac-Moody blocks on the torus and their monodromy representations. Using the free field representation developed by Bernard and Felder, an integral representation of the twisted two point spin-1/2-spin-1/2 conformal Kac-Moody blocks on the torus is computed. From this, an integral representation of the untwisted blocks is computed after careful removal of infinities. Finally, the untwisted blocks are used to get a representation of the Braid Group on the torus on two strings, in terms of quantum group q-numbers. (orig.).
Lie algebraic Noncommutative Gravity
Banerjee, R; Samanta, S; Banerjee, Rabin; Mukherjee, Pradip; Samanta, Saurav
2007-01-01
The minimal (unimodular) formulation of noncommutative general relativity, based on gauging the Poincare group, is extended to a general Lie algebra valued noncommutative structure. We exploit the Seiberg -- Witten map technique to formulate the theory as a perturbative Lagrangian theory. Detailed expressions of the Seiberg -- Witten maps for the gauge parameters, gauge potentials and the field strengths have been worked out. Our results demonstrate that notwithstanding the introduction of more general noncommutative structure there is no first order correction, exactly as happens for a canonical (i.e. constant) noncommutativity.
Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra
2001-01-01
The concept of a framed vertex operator algebra was studied in [DGH] (q-alg/9707008). This article is an analysis of all Virasoro frame stabilizers of the lattice VOA V for the E_8 root lattice, which is isomorphic to the E_8-level 1 affine Kac-Moody VOA V. We analyze the frame stabilizers, both as abstract groups and as subgroups of the Lie group Aut(V) = E_8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V). In particu...
Construct Weak Hopf Algebras by Using Borcherds Matrix
Institute of Scientific and Technical Information of China (English)
Zhi Xiang WU
2009-01-01
We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra y by adding a new generator J satisfying Jm = J for some integer m. We denote this algebra by wU τ q (y. This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra U q(y) of a generalized Kac-Moody algebra y.
Deciding isomorphism of Lie algebras
Graaf, W.A. de
2001-01-01
When doing calculations with Lie algebras one of the main problems is to decide whether two given Lie algebras are isomorphic. A partial solution to this problem is obtained by calculating structural invariants. There is also a direct method available which involves the computation of Grobner bases.
Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra
Lagraa, M
1997-01-01
We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.
Loop Virasoro Lie conformal algebra
Energy Technology Data Exchange (ETDEWEB)
Wu, Henan, E-mail: wuhenanby@163.com; Chen, Qiufan; Yue, Xiaoqing [Department of Mathematics, Tongji University, Shanghai 200092 (China)
2014-01-15
The Lie conformal algebra of loop Virasoro algebra, denoted by CW, is introduced in this paper. Explicitly, CW is a Lie conformal algebra with C[∂]-basis (L{sub i} | i∈Z) and λ-brackets [L{sub i} {sub λ} L{sub j}] = (−∂−2λ)L{sub i+j}. Then conformal derivations of CW are determined. Finally, rank one conformal modules and Z-graded free intermediate series modules over CW are classified.
Particle-like structure of Lie algebras
Vinogradov, A. M.
2017-07-01
If a Lie algebra structure 𝔤 on a vector space is the sum of a family of mutually compatible Lie algebra structures 𝔤i's, we say that 𝔤 is simply assembled from the 𝔤i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the 𝔤i's, one obtains a Lie algebra assembled in two steps from 𝔤i's, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled. We develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over R can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons.
Semiclassical states on Lie algebras
Energy Technology Data Exchange (ETDEWEB)
Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com [King’s College, 133 North River Street, Kingston, Pennsylvania 18702 (United States)
2015-03-15
The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.
Graded Lie Algebra Generating of Parastatistical Algebraic Relations
Institute of Scientific and Technical Information of China (English)
JING Si-Cong; YANG Wei-Min; LI Ping
2001-01-01
A new kind of graded Lie algebra (We call it Z2,2 graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the Z2,2 graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Lie groups, lie algebras, and representations an elementary introduction
Hall, Brian
2015-01-01
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compac...
An evaluation on Real Semisimple Lie Algebras
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
@@ The theory of Lie groups and Lie algebras stem from that of continuous groups founded by Sophus Lie at the end of 19th century. From the beginning, the theory of Lie groups and Lie algebras has displayed great value in both theoretical researches and applications.
Probability on real Lie algebras
Franz, Uwe
2016-01-01
This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.
DERIVATIONS AND EXTENSIONS OF LIE COLOR ALGEBRA
Institute of Scientific and Technical Information of China (English)
Zhang Qingcheng; Zhang Yongzheng
2008-01-01
In this article, the authors obtain some results concerning derivations of fi-nitely generated Lie color algebras and discuss the relation between skew derivation space SkDer(L) and central extension H2(L, F) on some Lie color algebras. Meanwhile, they generalize the notion of double extension to quadratic Lie color algebras, a sufficient con-dition for a quadratic Lie color algebra to be a double extension and further properties are given.
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-01
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, "Cohomologie des algèbres de Lie nilpotentes. Application à l'étude de la variété des algèbres de Lie nilpotentes," Bull. Soc. Math. France 98, 81-116 (1970)]. Also we give the dimension, using an adaptation of the {sl}(2,{C})-module Method, and a basis of such infinitesimal deformations in some generic cases.
On Nambu-Lie 3-algebra representations
Sochichiu, Corneliu
2008-01-01
We propose a recipe to construct matrix representations of Nambu--Lie 3-algebras in terms of irreducible representations of underlying Lie algebra. The case of Euclidean four-dimensional 3-algebra is considered in details. We find that representations of this 3-algebra are not possible in terms of only Hermitian matrices in spite of its Euclidean nature.
Symmetry via Lie algebra cohomology
Eastwood, Michael
2010-01-01
The Killing operator on a Riemannian manifold is a linear differential operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as arising from the identification of certain Lie algebra cohomologies. The point is that this case provides a model for more complicated operators similarly concerned with symmetry.
Leibniz algebras associated with representations of filiform Lie algebras
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
Computations in finite-dimensional Lie algebras
Cohen, A.M.; Graaf, W.A. de; Rónyai, L.
2001-01-01
This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System), within the computer algebra package GAP. A first sketch of the packagecan be found in Cohen and de Graaf[1]. Since then, in a collaborative
Computations in finite-dimensional Lie algebras
Directory of Open Access Journals (Sweden)
A. M. Cohen
1997-12-01
Full Text Available This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System, within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]. This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.
Post-Lie algebras and factorization theorems
Ebrahimi-Fard, Kurusch; Mencattini, Igor; Munthe-Kaas, Hans
2017-09-01
In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang-Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.
Additive Lie ($\\xi$-Lie) Derivations and Generalized Lie ($\\xi$-Lie) Derivations on Prime Algebras
Qi, Xiaofei
2010-01-01
The additive (generalized) $\\xi$-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumption, that an additive map $L$ is an additive (generalized) Lie derivation if and only if it is the sum of an additive (generalized) derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) $\\xi$-Lie derivation with $\\xi\
Killing Forms of Isotropic Lie Algebras
Malagon, Audrey
2010-01-01
This paper presents a method for computing the Killing form of an isotropic Lie algebra defined over an arbitrary field based on the Killing form of a subalgebra containing its anisotropic kernel. This approach allows for streamlined formulas for many Lie algebras of types E6 and E7 and yields a unified formula for all Lie algebras of inner type E6, including the anisotropic ones.
Gauge Theory of the Generalized Symmetry on the Torus Membrane
Institute of Scientific and Technical Information of China (English)
ZHAO WeiZhong; WANG Hong; ZHANG Jun
2001-01-01
The SDIFF(T2)local-generalized Kac-Moody G(T2) symmetry is an infinite-dimensional group on the torus membrane, whose Lie algebra is the semi-direct sum of the SDIFF(T2)local algebra and the generalized KacMoody algebra g(T2). In this paper, we construct the linearly realized gauge theory of the SDIFF(T2)loc1al-generalized Kac-Moody G(T2) symmetry.``
Constructions of Lie algebras and their modules
Seligman, George B
1988-01-01
This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. T...
Engel Subalgebras of n-Lie Algebras
Institute of Scientific and Technical Information of China (English)
Donald W. BARNES
2008-01-01
Engel subalgebras of finite-dimensional n Lie algebras are shown to have similar properties to those of Lie algebras.Using these,it is shown that an n Lie algebra,all of whose maximal subalgebras are ideals,is nilpotent.A primitive 2-soluble n Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate.A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel,provided that the field has su .ciently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.
Additive Lie (ζ-Lie) Derivations and Generalized Lie (ζ-Lie)Derivations on Prime Algebras
Institute of Scientific and Technical Information of China (English)
Xiao Fei QI; Jin Chuan HOU
2013-01-01
The additive (generalized) ζ-Lie derivations on prime algebras are characterized.It is shown,under some suitable assumptions,that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) ζ-Lie derivation with ζ ≠ 1 if and only if it is an additive (generalized) derivation satisfying L(ζA) =ζL(A) for all A.These results are then used to characterize additive (generalized) ζ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.
On the simplicity of Lie algebras associated to Leavitt algebras
Abrams, Gene
2009-01-01
For any field $K$ and integer $n\\geq 2$ we consider the Leavitt algebra $L = L_K(n)$. $L$ is an associative algebra, but we view $L$ as a Lie algebra using the bracket $[a,b]=ab-ba$ for $a,b \\in L$. We denote this Lie algebra as $L^-$, and consider its Lie subalgebra $[L^-,L^-]$. In our main result, we show that $[L^-,L^-]$ is a simple Lie algebra if and only if char$(K)$ divides $n-1$. For any positive integer $d$ we let $S = M_d(L_K(n))$ be the $d\\times d$ matrix algebra over $L_K(n)$. We give sufficient conditions for the simplicity and non-simplicity of the Lie algebra $[S^-,S^-]$.
Homology of Lie algebra of supersymmetries and of super Poincare Lie algebra
Energy Technology Data Exchange (ETDEWEB)
Movshev, M.V. [Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651 (United States); Schwarz, A., E-mail: schwarz@math.ucdavis.edu [Department of Mathematics, University of California, Davis, CA 95616 (United States); Xu, Renjun [Department of Physics, University of California, Davis, CA 95616 (United States)
2012-01-11
We study the homology and cohomology groups of super Lie algebras of supersymmetries and of super Poincare Lie algebras in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions {<=}11. For dimensions D=10,11 we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry Lie algebras.
A Class of Solvable Lie Algebras and Their Hom-Lie Algebra Structures
Institute of Scientific and Technical Information of China (English)
LI Xiao-chao; LI Dong-ya; JIN Quan-qin
2014-01-01
The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1 as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1. Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.
A twisted generalization of Lie-Yamaguti algebras
Gaparayi, Donatien
2010-01-01
A twisted generalization of Lie-Yamaguti algebras, called Hom-Lie-Yamaguti algebras, is defined. Hom-Lie-Yamaguti algebras generalize Hom-Lie triple systems (and susequently ternary Hom-Nambu algebras) and Hom-Lie algebras in the same way as Lie-Yamaguti algebras generalize Lie triple systems and Lie algebras. It is shown that the category of Hom-Lie-Yamaguti algebras is closed under twisting by self-morphisms. Constructions of Hom-Lie-Yamaguti algebras from classical Lie-Yamaguti algebras and Malcev algebras are given. It is observed that, when the ternary operation of a Hom-Lie-Yamaguti algebra expresses through its binary one in a specific way, then such a Hom-Lie-Yamaguti algebra is a Hom-Malcev algebra.
Matrix Lie Algebras and Integrable Couplings
Institute of Scientific and Technical Information of China (English)
ZHANG Yu-Feng; GUO Fu-Kui
2006-01-01
Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.
Noncommutative geometry with graded differential Lie algebras
Wulkenhaar, Raimar
1997-06-01
Starting with a Hilbert space endowed with a representation of a unitary Lie algebra and an action of a generalized Dirac operator, we develop a mathematical concept towards gauge field theories. This concept shares common features with the Connes-Lott prescription of noncommutative geometry, differs from that, however, by the implementation of unitary Lie algebras instead of associative * -algebras. The general scheme is presented in detail and is applied to functions ⊗ matrices.
Lie Admissible Non-Associative Algebras
Institute of Scientific and Technical Information of China (English)
H.Mohammad Ahmadi; Ki-Bong Nam; Jonathan Pakinathan
2005-01-01
A non-associative ring which contains a well-known associative ring or Lie ring is interesting. In this paper, a method to construct a Lie admissible non-associative ring is given; a class of simple non-associative algebras is obtained; all the derivations of the non-associative simple N0,0,1 algebra defined in this paper are determined; and finally, a solid algebra is defined.
The Lie Algebras in which Every Subspace s Its Subalgebra
Institute of Scientific and Technical Information of China (English)
WU MING-ZHONG
2009-01-01
In this paper, we study the Lie algebras in which every subspace is its subalgebra (denoted by HB Lie algebras). We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to g+Cidg, where g is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.
Classification and identification of Lie algebras
Snobl, Libor
2014-01-01
The purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The authors address the problem of expressing a Lie algebra obtained in some arbitrary basis in a more suitable basis in which all essential features of the Lie algebra are directly visible. This includes algorithms accomplishing decomposition into a direct sum, identification of the radical and the Levi decomposition, and the computation of the nilradical and of the Casimir invariants. Examples are given for each algorithm. For low-dimensional Lie algebras this makes it possible to identify the given Lie algebra completely. The authors provide a representative list of all Lie algebras of dimension less or equal to 6 together with their important properties, including their Casimir invariants. The list is ordered in a way to make identification easy, using only basis independent properties of the Lie algebras. They also describe certain cl...
Jardino, Sergio
2010-01-01
We extend the concept of a generalized Lie 3-algebra, known to octonions $\\mathbb{O}$, to split-octonions $\\mathbb{SO}$. In order to do that, we introduce a notational device that unifies the two elements product of both of the algebras. We have also proved that $\\mathbb{SO}$ is a Malcev algebra and have recalculated known relations for the structure constants in terms of the introduced structure tensor. An application of the split Lie $3-$algebra to a Bagger and Lambert gauge theory is also discussed.
Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations
Institute of Scientific and Technical Information of China (English)
Ling CHEN
2011-01-01
We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.
Central extension of graded Lie algebras
Welte, Angelika
2010-01-01
In this thesis we describe the universal central extension of two important classes of so-called root-graded Lie algebras defined over a commutative associative unital ring $k.$ Root-graded Lie algebras are Lie algebras which are graded by the root lattice of a locally finite root system and contain enough $\\mathfrak{sl}_2$-triples to separate the homogeneous spaces of the grading. Examples include the infinite rank analogs of the simple finite-dimensional complex Lie algebras. \\\\ In the thesis we show that in general the universal central extension of a root-graded Lie algebra $L$ is not root-graded anymore, but that we can measure quite easily how far it is away from being so, using the notion of degenerate sums, introduced by van der Kallen. We then concentrate on root-graded Lie algebras which are graded by the root systems of type $A$ with rank at least 2 and of type $C$. For them one can use the theory of Jordan algebras.
Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra
Institute of Scientific and Technical Information of China (English)
ZHANG Yu-Feng; LIU Jing
2008-01-01
By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 (2006) 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.
Constructing semisimple subalgebras of semisimple Lie algebras
de Graaf, Willem A
2010-01-01
Algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. The algorithms have been used to obtain classifications of the semisimple subalgebras of the simple Lie algebras of ranks <= 8. These have been made available as a database inside the SLA package of GAP4. The subalgebras in this database are explicitly given, as well as the inclusion relations among them.
SAYD modules over Lie-Hopf algebras
Rangipour, B
2011-01-01
In this paper a general van Est type isomorphism is established. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and SAYD modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is found at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes- Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate...
SAYD Modules over Lie-Hopf Algebras
Rangipour, Bahram; Sütlü, Serkan
2012-11-01
In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.
Sugawara construction and Casimir operators for Krichever-Novikov algebras
Schlichenmaier, M; Schlichenmaier, Martin; Sheinman, Oleg K
1995-01-01
We show how to obtain from highest weight representations of Krichever-Novikov algebras of affine type (also called higher genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations are given and Casimir operators are constructed. In an appendix the Sugawara construction for the multi-point situation is done.
COMPLETE LIE ALGEBRAS WITH l-STEP NILPOTENT RADICALS
Institute of Scientific and Technical Information of China (English)
高永存; 孟道冀
2002-01-01
The authors first give a necessary and sufficient condition for some solvable Lie algebras with l-step nilpotent radicals to be complete, and then construct a new class of infinite dimensional complete Lie algebras by using the modules of simple Lie algebras. The quotient algebras of this new constructed Lie algebras are non-solvable complete Lie algebras with l-step nilpotent radicals.
Lie algebras with given properties of subalgebras and elements
Zusmanovich, Pasha
2011-01-01
Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which each nonzero element is regular in the sense of Bourbaki), minimal nonabelian (i.e., nonabelian Lie algebras all whose proper subalgebras are abelian), and algebras of depth 2 (i.e., Lie algebras all whose proper subalgebras are abelian or minimal nonabelian).
Classification of filiform Lie algebras of order 3
Navarro, Rosa María
2016-12-01
Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne's result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.
Simple Lie algebras arising from Leavitt path algebras
Abrams, Gene
2011-01-01
For a field K and directed graph E, we analyze those elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E), L_K(E)]. This analysis allows us to give easily computable necessary and sufficient conditions to determine which Lie algebras of the form [L_K(E), L_K(E)] are simple, when E is row-finite and L_K(E) is simple.
Lie algebras and degenerate Affine Hecke Algebras of type A
Arakawa, T
1997-01-01
We construct a family of exact functors from the BGG category of representations of the Lie algebra sl to the category of finite-dimensional representations of the degenerate (or graded) affine Hecke algebra H of GL. These functors transform Verma modules to standard modules or zero, and simple modules to simple modules or zero. Any simple H-module can be thus obtained.
Sati, Hisham
2015-01-01
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane...
Lie Subalgebras in a Certain Operator Lie Algebra with Involution
Institute of Scientific and Technical Information of China (English)
Shan Li SUN; Xue Feng MA
2011-01-01
We show in a certain Lie'-algebra,the connections between the Lie subalgebra G+:＝G+G*+[G,G*],generated by a Lie subalgebra G,and the properties of G.This allows us to investigate some useful information about the structure of such two Lie subalgebras.Some results on the relations between the two Lie subalgebras are obtained.As an application,we get the following conclusion:Let A (∪) B(X)be a space of self-adjoint operators and L:＝A ⊕ iA the corresponding complex Lie*-algebra.G+＝G+G*+[G,G*]and G are two LM-decomposable Lie subalgebras of,L with the decomposition G+＝R(G+)+S,G＝RG+SG,and RG (∪) R(C+).Then G+ is ideally finite iff RG+:＝RG+RG*+[RG,RG*]is a quasisolvable Lie subalgebra,SG+:＝SG+SG*+[SG,SG*]is an ideally finite semisimple Lie subalgebra,and [RG,SG]＝[RG*,SG]＝{0}.
Dimension of the $c$-nilpotent multiplier of Lie algebras
Indian Academy of Sciences (India)
MEHDI ARASKHAN; MOHAMMAD REZA RISMANCHIAN
2016-08-01
The purpose of this paper is to derive some inequalities for dimension of the $c$-nilpotent multiplier of finite dimensional Lie algebras and their factor Lie algebras. We further obtain an inequality between dimensions of $c$-nilpotent multiplier of Lie algebra $L$ and tensor product of a central ideal by its abelianized factor Lie algebra
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
Hijligenberg, N.W. van den; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
N.W. van den Hijligenberg; R. Martini
1995-01-01
textabstractWe discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra
Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra
van den Hijligenberg, N.W.; van den Hijligenberg, N.W.; Martini, Ruud
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of
Differential Hopf algebra structures on the universal enveloping algebra of a lie algebra
Hijligenberg, van den, N.W.; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g)$. The construction of such differential structures is interpreted in terms of colour Lie superalgebras.
Riemannian manifolds as Lie-Rinehart algebras
Pessers, Victor; van der Veken, Joeri
2016-07-01
In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.
Spiders for rank 2 Lie algebras
Kuperberg, G
1996-01-01
A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.
Transformation groups and Lie algebras
Ibragimov, Nail H
2013-01-01
This book is based on the extensive experience of teaching for mathematics, physics and engineering students in Russia, USA, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics. The methods of local Lie groups discussed in the book provide universal and effective method for solving nonlinear differential equations analytically. Introduction to approximate transformation groups also contained in the book helps to develop skills in constructing approximate solutions for differential equations with a small parameter.
Lie algebras and linear differential equations.
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Differential Hopf algebra structures on the Universal Enveloping Algebra of a Lie Algebra
van den Hijligenberg, N.W.; van den Hijligenberg, N.; Martini, Ruud
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincaré–Birkhoff–Witt type on the universal enveloping algebra of a Lie algebra g. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebrastructure of U(g).
A Local Characterization of Lie Homomorphisms of Nest Algebras
Institute of Scientific and Technical Information of China (English)
YANG Miao-xia; ZHANG Jian-hua
2014-01-01
In this paper, linear maps preserving Lie products at zero points on nest algebras are studied. It is proved that every linear map preserving Lie products at zero points on any finite nest algebra is a Lie homomorphism. As an application, the form of a linear bijection preserving Lie products at zero points between two finite nest algebras is obtained.
NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sln(fCq) WITH NULLITY M
Institute of Scientific and Technical Information of China (English)
Jie TONG; Quanqin JIN
2013-01-01
Non-commutative Poisson algebras are the algebras having both an associa-tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(fCq) are determined.
A-扩张Lie Rinehart代数%On the A-extended Lie Rinehart Algebras
Institute of Scientific and Technical Information of China (English)
陈酌; 祁玉海
2007-01-01
The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary,commutative, associative algebra A. It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group, analogous to the well known relationship of Lie algebras and Lie groups.
Lie algebra contractions and separation of variables
Vinternits, P; Pogosyan, G S; Sissakian, A N
2001-01-01
The concept of analytical Lie group contractions is introduced to relate the separation of variables in space of constant nonzero curvature to separation in Euclidean or pseudo-Euclidean spaces. The contraction parameter is introduced explicitly into the basis of the Lie algebra, the Laplace-Beltrami operator, the complete set of commuting operators, the coordinates themselves and into the solutions. This enables to obtain asymptotic formulae connecting special functions related to the groups O(n) and O(n,1) to those related to Euclidean and pseudo-Euclidean groups
Quantum Lie algebras of type A$_{n}$
Sudbery, A I
1995-01-01
It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's. This gives rise to Lie algebra-like generators and relations for the locally finite part of the quantised enveloping algebra, and suggests a canonical Poincare-Birkhoff-Witt basis.
Central Extension for the Triangular Derivation Lie Algebra
Institute of Scientific and Technical Information of China (English)
Chunming LI; Ping XU
2012-01-01
In this paper,we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus,which are called the Triangular derivation Lie algebra.We give the structure and the central extension of Triangular derivation Lie algebra.
Dobrev, V K
2013-01-01
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of {\\it parabolic relation} between two non-compact semisimple Lie algebras g and g' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E_{7(7)} which is parabolically related to the CLA E_{7(-25)}, the parabolic subalgebras including E_{6(6)} and E_{6(-6)} . Other interesting examples are the orthogonal algebras so(p,q) all of which are parabolically related to the conformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including the Lorentz subalgebra so(n-1,1) and its analogs so(p-1,...
Homomorphisms between JC*-algebras and Lie C*-algebras
Institute of Scientific and Technical Information of China (English)
Chun Gil PARK; Jin Chuan HOU; Sei Qwon OH
2005-01-01
It is shown that every almost *-homomorphism h: A → B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r ＞ 1) for all x ∈ A, and that every almost linear mapping h: A → B is a *-homomorphism when h(2nu o y) = h(2nu) o h(y),h(3nu o y) = h(3nu) o h(y) or h(qnu o y) = h(qnu) o h(y) for all unitaries u ∈ A, all y ∈ A, and n = 0, 1, Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings.We prove that every almost *-homomorphism h: A → B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r ＞ 1) for all x ∈ A.
Automorphisms of Strong Homotopy Lie Algebras of Local Observables
Ritter, Patricia
2015-01-01
There is a well-established procedure of assigning a strong homotopy Lie algebra of local observables to a multisymplectic manifold which can be regarded as part of a categorified Poisson structure. For a 2-plectic manifold, the resulting Lie 2-algebra is isomorphic to a sub Lie 2-algebra of a natural Lie 2-algebra structure on an exact Courant algebroid. We generalize this statement to arbitrary n-plectic manifolds and study automorphisms on the arising Lie n-algebras. Our observations may be useful in studying the quantization problem on multisymplectic manifolds.
Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, Rutwig [Dpto. Geometria y Topologia, Fac CC Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, E-28040 Madrid (Spain)
2006-03-10
Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie algebras Iu(p,q). This procedure is extended to contractions of Iu(p,q) isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Lie algebras Isu(p-1,q), providing an additional analytical method to obtain their invariants. Further, matrix formulae for the invariants of other inhomogeneous Lie algebras are presented.
Restricted and quasi-toral restricted Lie-Rinehart algebras
Directory of Open Access Journals (Sweden)
Sun Bing
2015-09-01
Full Text Available In this paper, we introduce the definition of restrictable Lie-Rinehart algebras, the concept of restrictability is by far more tractable than that of a restricted Lie-Rinehart algebra. Moreover, we obtain some properties of p-mappings and restrictable Lie-Rinehart algebras. Finally, we give some sufficient conditions for the commutativity of quasi-toral restricted Lie-Rinehart algebras and study how a quasi-toral restricted Lie-Rinehart algebra with zero center and of minimal dimension should be.
A Specialization of Prinjective Ringel-Hall Algebra and the associated Lie algebra
Institute of Scientific and Technical Information of China (English)
Justyna KOSAKOWSKA
2008-01-01
In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type,by generators and relations.This gives us a generalisation of Serre relations for semisimple Lie algebras.Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.
Construction of Lie algebras and invariant tensors through abelian semigroups
Energy Technology Data Exchange (ETDEWEB)
Izaurieta, Fernando; RodrIguez, Eduardo; Salgado, Patricio [Departamento de Fisica, Universidad de Concepcion, Casilla 160-C, Concepcion (Chile)], E-mail: fizaurie@gmail.com, E-mail: edurodriguez@udec.cl, E-mail: pasalgad@udec.cl
2008-11-01
The Abelian Semigroup Expansion Method for Lie Algebras is briefly explained. Given a Lie Algebra and a discrete abelian semigroup, the method allows us to directly build new Lie Algebras with their corresponding non-trivial invariant tensors. The Method is especially interesting in the context of M-Theory, because it allows us to construct M-Algebra Invariant Chern-Simons/Transgression Lagrangians in d = 11.
Lie algebraic noncommuting structures from reparametrisation symmetry
Gangopadhyay, S
2007-01-01
We extend our earlier work of revealing both space-space and space-time noncommuting structures in various models in particle mechanics exhibiting reparametrisation symmetry. We show explicitly (in contrast to the earlier results in our paper \\cite{sg}) that for some special choices of the reparametrisation parameter $\\epsilon$, one can obtain space-space noncommuting structures which are Lie-algebraic in form even in the case of the relativistic free particle. The connection of these structures with the existing models in the literature is also briefly discussed. Further, there exists some values of $\\epsilon$ for which the noncommutativity in the space-space sector can be made to vanish. As a matter of internal consistency of our approach, we also study the angular momentum algebra in details.
The graded Lie algebra of general relativity
Reiterer, Michael
2014-01-01
We construct a graded Lie algebra in which a solution to the vacuum Einstein equations is any element of degree 1 whose bracket with itself is zero. Each solution generates a cochain complex, whose first cohomology is linearized gravity about that solution. We gauge-fix to get a smaller cochain complex with the same cohomologies (deformation retraction). The new complex is much smaller, it consists of the solution spaces of linear homogeneous wave equations (symmetric hyperbolic equations). The algorithm that produces these gauges and wave equations is both for linearized gravity and the full Einstein equations. The gauge groupoid is the groupoid of rank 2 complex vector bundles.
New phases of D ge 2 current and diffeomorphism algebras in particle physics
Energy Technology Data Exchange (ETDEWEB)
Tze, Chia-Hsiung.
1990-09-01
We survey some global results and open issues of current algebras and their canonical field theoretical realization in D {ge} 2 dimensional spacetime. We assess the status of the representation theory of their generalized Kac-Moody and diffeomorphism algebras. Particular emphasis is put on higher dimensional analogs of fermi-bose correspondence, complex analyticity and the phase entanglements of anyonic solitons with exotic spin and statistics. 101 refs.
Free-field realisations of BMS$_3$ and super-BMS$_3$ algebras
Banerjee, Nabamita; Mukhi, Sunil; Neogi, Turmoli
2015-01-01
We construct an explicit realisation of the BMS$_3$ algebra with nonzero central charges using holomorphic free fields. This can be extended by the addition of chiral matter to a realisation having arbitrary values for the two independent central charges. We show that our construction naturally extends to a coupled SU(2)-BMS$_3$ system where the SU(2) Kac-Moody symmetry is realised via the Wakimoto representation, and to the supersymmetric BMS$_3$ algebra.
Construction of the elliptic Gaudin system based on Lie algebra
Institute of Scientific and Technical Information of China (English)
CAO Li-ke; LIANG Hong; PENG Dan-tao; YANG Tao; YUE Rui-hong
2007-01-01
Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics.The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra.Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, An, Bn, Cn, Dn, and we calculate a classical r-matrix for the elliptic Gaudin system with spin.
$W_{\\infty}$ algebra in the integer quantum Hall effects
Azuma, Hiroo
1994-01-01
We investigate the $W_{\\infty}$ algebra in the integer quantum Hall effects. Defining the simplest vacuum, the Dirac sea, we evaluate the central extension for this algebra. A new algebra which contains the central extension is called the $W_{1+\\infty}$ algebra. We show that this $W_{1+\\infty}$ algebra is an origin of the Kac-Moody algebra which determines the behavior of edge states of the system. We discuss the relation between the $W_{1+\\infty}$ algebra and the incompressibility of the int...
Fermionic realisations of simple Lie algebras
de Azcárraga, J A
2000-01-01
We study the representation ${\\cal D}$ of a simple compact Lie algebra $\\g$ of rank l constructed with the aid of the hermitian Dirac matrices of a (${\\rm dim} \\g$)-dimensional euclidean space. The irreducible representations of $\\g$ contained in ${\\cal D}$ are found by providing a general construction on suitable fermionic Fock spaces. We give full details not only for the simplest odd and even cases, namely su(2) and su(3), but also for the next (${dim} \\g$)-even case of su(5). Our results are far reaching: they apply to any $\\g$-invariant quantum mechanical system containing ${\\rm dim} \\g$ fermions. Another reason for undertaking this study is to examine the role of the $\\g$-invariant fermionic operators that naturally arise. These are given in terms of products of an odd number of gamma matrices, and include, besides a cubic operator, (l-1) fermionic scalars of higher order. The latter are constructed from the Lie algebra cohomology cocycles, and must be considered to be of theoretical significance simila...
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras
Energy Technology Data Exchange (ETDEWEB)
Ammar, F [Faculte des Sciences, Universite de Sfax, BP 1171, 3000 Sfax (Tunisia); Makhlouf, A [Laboratoire de Mathematiques, Informatique et Applications, Universite de Haute Alsace, 4, rue des Freres Lumiere F-68093 Mulhouse (France); Silvestrov, S, E-mail: Faouzi.Ammar@rnn.fss.t, E-mail: Abdenacer.Makhlouf@uha.f, E-mail: sergei.silvestrov@math.lth.s [Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund (Sweden)
2010-07-02
In this paper we construct ternary q-Virasoro-Witt algebras which q-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary q-Virasoro-Witt algebras constructed in this paper are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield a Hom-Nambu-Lie algebra structure for q-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro-Witt algebras we construct, carry a structure of the ternary Hom-Nambu-Lie algebra for all values of the involved parameters.
Test Rank of an Abelian Product of a Free Lie Algebra and a Free Abelian Lie Algebra
Indian Academy of Sciences (India)
Naime Ekici; Nazar Şahin Öğüşlü
2011-08-01
Let be a free Lie algebra of rank ≥ 2 and be a free abelian Lie algebra of rank ≥ 2. We prove that the test rank of the abelian product $F× A$ is . Morever we compute the test rank of the algebra $F/ k(F)'$.
Universal representations of Lie algebras by coderivations
Petracci, Emanuela
2003-01-01
A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this theorem to a class of nilpotent Lie superalgebras. Other applications are presented. Our results are new already for Lie algebras.
Lie symmetry algebra of one-dimensional nonconservative dynamical systems
Institute of Scientific and Technical Information of China (English)
Liu Cui-Mei; Wu Run-Heng; Fu Jing-Li
2007-01-01
Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1-1 mapping,the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
Directory of Open Access Journals (Sweden)
Evgeny Feigin
2008-10-01
Full Text Available Let L be the basic (level one vacuum representation of the affine Kac-Moody Lie algebra ^g. The m-th space F_m of the PBW filtration on L is a linear span of vectors of the form x_1dots x_lv_0, where l ≤ m, x_i in ^g and v_0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space L^{gr} with respect to the PBW filtration. The ''top-down'' description deals with a structure of L^{gr} as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field e_θ(z2, which corresponds to the longest root θ. The ''bottom-up'' description deals with the structure of L^{gr} as a representation of the current algebra g otimes C[t]. We prove that each quotient F_m/F_{m-1} can be filtered by graded deformations of the tensor products of m copies of g.
A new kind of graded Lie algebra and parastatistical supersymmetry
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
In this paper the usual Z2 graded Lie algebra is generalized to a new form, which may be called Z2,2 graded Lie algebra. It is shown that there exist close connections between the Z2,2 graded Lie algebra and parastatistics, so the Z2,2 can be used to study and analyse various symmetries and supersymmetries of the paraparticle systems.
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
Kersten, Paul H.M.; Gragert, Peter K.H.
1983-01-01
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.
The structure of split regular BiHom-Lie algebras
Calderón, Antonio J.; Sánchez, José M.
2016-12-01
We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U +∑jIj with U a linear subspace of a fixed maximal abelian subalgebra H and any Ij a well described (split) ideal of L, satisfying [Ij ,Ik ] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.
Non-filiform Characteristically Nilpotent and Complete Lie Algebras
Institute of Scientific and Technical Information of China (English)
José María Ancochea-Bermúdez; Rutwig Campoamor
2002-01-01
In this paper, we construct large families of characteristically nilpotent Lie algebras by considering deformations of the Lie algebra (g)4(m,m-1) of type Qn,which arises as a central naturally graded extension of the filiform Lie algebra Ln.By studying the graded cohomology spaces, we obtain that the sill algebras associated to the models (g)4(m,m-1) can be interpreted as nilradicals of solvable, complete Lie algebras. For extreme cocycles, we obtain moreover nilradicals of rigid laws.By considering supplementary cocycles, we construct, for any dimension n (＞－) 9,non-filiform characteristically nilpotent Lie algebras with mixed characteristic sequence and show that for certain deformations, these deformations are compatible with central extensions.
Holomorph of Lie color algebras%Lie color代数的全形
Institute of Scientific and Technical Information of China (English)
杨恒云
2007-01-01
给出Lie color代数全形的一些性质,证明Lie color代数L的全形有分解(H)(L)=L(+)Z(H)(L)(L)的充分必要条件是它是完备Lie color代数.%To the holomorph of Lie color algebras, some properties are studied. A Lie color algebra L is complete if and only if (H)(L) = L(+)Z(H)(L) (L).
Lie algebra type noncommutative phase spaces are Hopf algebroids
Meljanac, Stjepan; Škoda, Zoran; Stojić, Martina
2016-11-01
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.
Lie algebra type noncommutative phase spaces are Hopf algebroids
Meljanac, Stjepan
2014-01-01
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.
Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W (2, 2) Algebra
Institute of Scientific and Technical Information of China (English)
La Mei YUAN; Hong YOU
2014-01-01
This paper aims to study low dimensional cohomology of Hom-Lie algebras and the q-deformed W (2, 2) algebra. We show that the q-deformed W (2, 2) algebra is a Hom-Lie algebra. Also, we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the sec-ond cohomology group of the q-deformed W (2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras. As application, we compute all α k-derivations and in particular the first cohomology group of the q-deformed W (2, 2) algebra.
Symmetry algebras in Chern-Simons theories with boundary: canonical approach
Energy Technology Data Exchange (ETDEWEB)
Park, Mu-In. E-mail: mipark@physics.sogang.ac.kr
1999-04-05
I consider the classical Kac-Moody algebra and Virasoro algebra in Chern-Simons theory with boundary within Dirac's canonical method and Noether's procedure. It is shown that the usual (bulk) Gauss law constraint becomes a second-class constraint because of the boundary effect. From this fact, the Dirac bracket can be constructed explicitly without introducing additional gauge conditions and the classical Kac-Moody and Virasoro algebras are obtained within the usual Dirac method. The equivalence to the symplectic reduction method is presented and the connection to the Banados' work is clarified. Also the generalization to the Yang-Mills-Chern-Simons theory is considered where the diffeomorphism symmetry is broken by the (three-dimensional) Yang-Mills term. In this case, the same Kac-Moody algebras are obtained although the two theories are sharply different in the canonical structures. Both models realize the holography principle explicitly and the pure CS theory reveals the correspondence of the Chern-Simons theory with boundary/conformal field theory, which is more fundamental and generalizes the conjectured anti-de Sitter/conformal field theory correspondence.
Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups
Batat, Wafaa
2011-01-01
We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are sol-solitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons.
Structures of W(2.2 Lie conformal algebra
Directory of Open Access Journals (Sweden)
Yuan Lamei
2016-01-01
. In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.
Computations with reachable elements in simple Lie algebras
de Graaf, Willem
2010-01-01
We report on some computations with reachable elements in simple Lie algebras of exceptional type within the SLA package of GAP4. These computations confirm the classification of such elements by Elashvili and Grelaud. Secondly they answer a question from Panyushev. Thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types.
Graded Automorphism Group of TKK Lie Algebra over Semilattice
Institute of Scientific and Technical Information of China (English)
Zhang Sheng XIA
2011-01-01
Every extended affine Lie algebra of type A1 and nullity v with extended affine root system R(A1, S), where S is a semilattice in Rv, can be constructed from a TKK Lie algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction. In this article we consider the Zn-graded automorphism group of the TKK Lie algebra T(J(S)), where S is the "smallest" semilattice in Euclidean space Rn.
On Split Lie Algebras with Symmetric Root Systems
Indian Academy of Sciences (India)
Antonio J Calderón Martín
2008-08-01
We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such algebras is of the form $L=\\mathcal{U}+\\sum_j I_j$ with $\\mathcal{U}$ a subspace of the abelian Lie algebra and any $I_j$ a well described ideal of , satisfying $[I_j,I_k]=0$ if $j≠ k$. Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.
Boolean-Lie algebras and the Leibniz rule
Energy Technology Data Exchange (ETDEWEB)
Bazso, Fueloep [KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences, PO Box 49, H-1525 Budapest (Hungary); Labos, Elemer [Neurobiology Research Group, United Research Organization of the Hungarian Academy of Sciences and Semmelweis University, H-1450 Budapest, PO Box 95 (Hungary)
2006-06-02
Using internal negations acting on Boolean functions, the notion of Boolean-Lie algebra is introduced. The underlying Lie product is the Boolean analogue of the Poisson bracket. The structure of a Boolean-Lie algebra is determined; it turns out to be solvable, but not nilpotent. We prove that the adjoint representation of an element of the Boolean-Lie algebra acts as a derivative operator on the space of Boolean functions. The adjoint representation is related to the previously known concept of the sensitivity function. Using the notion of adjoint representation we give the definition of a temporal derivative applicable to iterative dynamics of Boolean mappings.
Leibniz algebras associated with some finite-dimensional representation of Diamond Lie algebra
Camacho, Luisa M.; Ladra, Manuel; Karimjanov, Iqboljon A.; Omirov, Bakhrom A.
2016-03-01
In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un 1, Un 2 or Wn 1 or Wn 2.
Relativity symmetries and Lie algebra contractions
Energy Technology Data Exchange (ETDEWEB)
Cho, Dai-Ning; Kong, Otto C.W., E-mail: otto@phy.ncu.edu.tw
2014-12-15
We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics.
Simplicities and Automorphisms of a Sp ecial Infinite Dimensional Lie Algebra
Institute of Scientific and Technical Information of China (English)
YU De-min; LI Ai-hua
2013-01-01
In this paper, a special infinite dimensional Lie algebra is studied. The infinite dimensional Lie algebra appears in the fields of conformal theory, mathematical physics, statistic mechanics and Hamilton operator. The infinite dimensional Lie algebras is pop-ularized Virasoro-like Lie algebra. Isomorphisms, homomorphisms, ideals of the infinite dimensional Lie algebra are studied.
The index of centralizers of elements of reductive Lie algebras
Charbonnel, Jean-Yves
2010-01-01
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
Non-commutative Poisson Algebra Structures on the Lie Algebra son(CQ)
Institute of Scientific and Technical Information of China (English)
Jie Tong; Quanqin Jin
2007-01-01
Non-commutative Poisson algebras are the algebras having both an associativealgebra structure and a Lie algebra structure together with the Leibniz law.In this paper,the non-commutative poisson algebra structures on son(CQ) are determined.
Freely generated vertex algebras and non-linear Lie conformal algebras
De Sole, Alberto; Kac, Victor
2003-01-01
We introduce the notion of a non--linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a non--linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.
Solvable Lie algebras with naturally graded nilradicals and their invariants
Energy Technology Data Exchange (ETDEWEB)
Ancochea, J M; Campoamor-Stursberg, R; Vergnolle, L Garcia [Departamento GeometrIa y TopologIa, Fac. CC. Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, E-28040 Madrid (Spain)
2006-02-10
The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analysed, and their generalized Casimir invariants are calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension 2n - 1, indicating that gauge theories (with ghosts) are possible on these subalgebras.
Internal labelling operators and contractions of Lie algebras
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, R [Dpto. GeometrIa y TopologIa, Fac. CC. Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, E-28040 Madrid (Spain)
2007-12-07
We analyse under which conditions the missing label problem associated with a reduction chain s' subset of s of (simple) Lie algebras can be completely solved by means of an Inoenue-Wigner contraction g naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labelling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions of semi-simple algebras.
q-deformed Lie algebras and fractional calculus
Herrmann, Richard
2007-01-01
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. For the fractional harmonic oscillator, the corresponding q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool e.g. to describe the ground state spectra of even-even nuclei. In addition, the equivalence of rotational and vibrational spectra for fractional q-deformed Lie algebras is shown and the $B_\\alpha(E2)$ values for the fractional q-deformed symmetric rotor are calculated.
3D Object Recognition Based on Linear Lie Algebra Model
Institute of Scientific and Technical Information of China (English)
LI Fang-xing; WU Ping-dong; SUN Hua-fei; PENG Lin-yu
2009-01-01
A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed.Then an algorithm of 3D object recognition using the linear Lie algebra models is presented.It is a convenient recognition method for the objects which are symmetric about some axis.By using the presented algorithm,the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained.At last some recognition results of practicalities are given.
Fricke Lie algebras and the genus zero property in Moonshine
Carnahan, Scott
2017-10-01
We give a new, simpler proof that the canonical actions of finite groups on Fricke-type Monstrous Lie algebras yield genus zero functions in generalized Monstrous Moonshine, using a Borcherds–Kac–Moody Lie algebra decomposition due to Jurisich. We describe a compatibility condition, arising from the no-ghost theorem in bosonic string theory, that yields the genus zero property. We give evidence for and against the conjecture that such a compatibility for symmetries of the Monster Lie algebra gives a characterization of the Monster group.
Lie algebras determined by finite valued Auslander-Reiten quivers
Institute of Scientific and Technical Information of China (English)
张顺华
1997-01-01
Let r denote a connected valued Auslander-Reiten quiver,let (Γ) denote the free abelian group generated by the vertex set Γ0 and let Γ be the universal cover of Γ with fundamental group G.It is proved that when Γ is a finite connected valued Auslander-Reiten quiver,(Γ) is a Lie subalgebra of (Γ) and is just the "rbit" Lie algebra (Γ)/G,where (Γ)1 is the degenerate Hall algebra of Γ and (Γ)/G is the "orbit" Lie algebra induced by Γ.
Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras
Ondrus, Matthew; Wiesner, Emilie
2016-07-01
This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
Noncommutative Gravity and the *-Lie algebra of diffeomorphisms
Aschieri, P
2007-01-01
We construct functions and tensors on noncommutative spacetime by systematically twisting the corresponding commutative structures. The study of the deformed diffeomorphisms (and Poincare) Lie algebra allows to construct a noncomutative theory of gravity.
Noncommutative Gravity and the *-Lie algebra of diffeomorphisms
Aschieri, Paolo
2008-07-01
We construct functions and tensors on noncommutative spacetime by systematically twisting the corresponding commutative structures. The study of the deformed diffeomorphisms (and Poincaré) Lie algebra allows to construct a noncomutative theory of gravity.
Calculations on Lie Algebra of the Group of Affine Symplectomorphisms
Directory of Open Access Journals (Sweden)
Zuhier Altawallbeh
2017-01-01
Full Text Available We find the image of the affine symplectic Lie algebra gn from the Leibniz homology HL⁎(gn to the Lie algebra homology H⁎Lie(gn. The result shows that the image is the exterior algebra ∧⁎(wn generated by the forms wn=∑i=1n(∂/∂xi∧∂/∂yi. Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology HH⁎-1(U(gn is isomorphic to H⁎-1Lie(gn,U(gnad. Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.
Monomial bases for free pre-Lie algebras
al-Kaabi, Mahdi Jasim Hasan
2013-01-01
In this paper, we study the concept of free pre-Lie algebra generated by a (non-empty) set. We review the construction of A. Agrachev and R. Gamkrelidze of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial basis vectors in terms of the rooted trees basis exhibited by F. Chapoton and M. Livernet. We also show that this matrix is unipotent and we find an explicit expression for its coefficients.
The Virasoro Algebra and Some Exceptional Lie and Finite Groups
Directory of Open Access Journals (Sweden)
Michael P. Tuite
2007-01-01
Full Text Available We describe a number of relationships between properties of the vacuum Verma module of a Virasoro algebra and the automorphism group of certain vertex operator algebras. These groups include the Deligne exceptional series of simple Lie groups and some exceptional finite simple groups including the Monster and Baby Monster.
Invariants of solvable rigid Lie algebras up to dimension 8
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, Rutwig [Depto Geometria y Topologia, Fac. CC Matematicas UCM, Madrid (Spain)]. E-mail: rutwig@nfssrv.mat.ucm.es
2002-08-02
The invariants of all complex solvable rigid Lie algebras up to dimension 8 are computed. Moreover we show, for rank 1 solvable algebras, some criteria to deduce the non-existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the Casimir invariants of the associated nilradical. (author)
Connes-Moscovici characteristic map is a Lie algebra morphism
Menichi, Luc
2010-01-01
Let $H$ be a Hopf algebra with a modular pair in involution $(\\Character,1)$. Let $A$ be a (module) algebra over $H$ equipped with a non-degenerated $\\Character$-invariant 1-trace $\\tau$. We show that Connes-Moscovici characteristic map $\\varphi_\\tau:HC^*_{(\\Character,1)}(H)\\to HC^*_\\lambda(A)$ is a morphism of graded Lie algebras. We also have a morphism $\\Phi$ of Batalin-Vilkovisky algebras from the cotorsion product of $H$, $\\text{Cotor}_H^*({\\Bbbk},{\\Bbbk})$, to the Hochschild cohomology of $A$, $HH^*(A,A)$. Let $K$ be both a Hopf algebra equipped with a modular pair in involution $(1,u)$ and a symmetric Frobenius algebra. Then this morphism of Batalin-Vilkovisky algebras $\\Phi:\\text{Cotor}_{K^\\vee}^*(\\mathbb{F},\\mathbb{F})\\cong \\text{Ext}_{K}(\\mathbb{F},\\mathbb{F}) \\hookrightarrow HH^*(K,K)$ is injective.
Inverse Limits in Representations of a Restricted Lie Algebra
Institute of Scientific and Technical Information of China (English)
Yu Feng YAO; Bin SHU; Yi Yang LI
2012-01-01
Let (g,[p]) be a restricted Lie algebra over an algebraically closed field of characteristic p ＞ 0.Then the inverse limits of "higher" reduced enveloping algebras {uxs (g) | s ∈ N} with x running over g* make representations of g split into different "blocks".In this paper,we study such an infinitedimensional algebra (A)x(g):=lim← Uxs (g) for a given x ∈ g*.A module category equivalence is built between subcategories of U(g)-mod and (A)x(g)-mod.In the case of reductive Lie algebras,(quasi)generalized baby Verma modules and their properties are described.Furthermore,the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized x-reduced module category are precisely determined,and a higher reciprocity in the case of regular nilpotent is obtained,generalizing the ordinary reciprocity.
Developments and retrospectives in Lie theory algebraic methods
Penkov, Ivan; Wolf, Joseph
2014-01-01
This volume reviews and updates a prominent series of workshops in representation/Lie theory, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, and mathematical physics. Many of the contributors have had leading roles in both the classical and modern developments of Lie theory and its applications. This Work, entitled Developments and Retrospectives in Lie Theory, and comprising 26 articles, is organized in two volumes: Algebraic Methods and Geometric and Analytic Methods. This is the Algebraic Methods volume. The Lie Theory Workshop series, founded by Joe Wolf and Ivan Penkov and joined shortly thereafter by Geoff Mason, has been running for over two decades. Travel to the workshops has usually been supported by the NSF, and local universities have provided hospitality. The workshop talks have been seminal in describing new perspectives in the field covering broad areas of current research. Mos...
On squares of representations of compact Lie algebras
Energy Technology Data Exchange (ETDEWEB)
Zeier, Robert, E-mail: robert.zeier@ch.tum.de [Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching (Germany); Zimborás, Zoltán, E-mail: zimboras@gmail.com [Department of Computer Science, University College London, Gower St., London WC1E 6BT (United Kingdom)
2015-08-15
We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.
Lie algebras for some specific dissipative Landau–Zener problems
Energy Technology Data Exchange (ETDEWEB)
Kenmoe, M.B. [Mesoscopic and Multilayer Structures Laboratory (MMSL), Faculty of Science, Department of Physics, University of Dschang (Cameroon); Mkam Tchouobiap, S.E., E-mail: esmkam@yahoo.com [Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS), Department of Physics, Faculty of Sciences, University of Buea, PO Box 63, Buea (Cameroon); Danga, J.E.; Kenfack Sadem, C.; Fai, L.C. [Mesoscopic and Multilayer Structures Laboratory (MMSL), Faculty of Science, Department of Physics, University of Dschang (Cameroon)
2015-03-20
We demonstrate that some specific problems of Landau–Zener transitions in a qubit coupled to an environment (problems designed as dissipative) can be matched onto the frame of the original problem without dissipation, providing an appropriate Lie algebra. Focusing on the origin of quantum noises, the cases of bosonic and spin baths are considered and presented. Finally, making use of the algebra framework, the logic is shown in action for respectively two important additional quantum models, namely the Jaynes–Cummings and an isolated double quantum dots models. - Highlights: • A finite temperature result for dissipative Landau–Zener transitions in a qubit coupled to an environment is proposed. • The quantum noises for bosonic and spin baths are considered. • Lie algebras reduction method coupled to the separation method and the fast driving approximation is proposed. • Jaynes–Cummings and a double quantum dots models are studied as illustrations of the algebra.
Classical Mechanics on Noncommutative Space with Lie-algebraic Structure
Miao, Yan-Gang; Yu, Shao-Jie
2009-01-01
We investigate the kinetics of a particle exerted by a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two general sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle by means of the Hamiltonian formalism defined on a Poisson manifold. Our results {\\em not only} include that of a recent work as our special cases, {\\em but also} provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable $t\\dot{x}$-, $\\dot{(xx)}$-, and $\\ddot{(xx)}$-dependence besides with the usual $t$-, $x$-, and $\\dot{x}$-dependence, originating...
Renormalization group flows and continual Lie algebras
Bakas, Ioannis
2003-01-01
We study the renormalization group flows of two-dimensional metrics in sigma models and demonstrate that they provide a continual analogue of the Toda field equations based on the infinite dimensional algebra G(d/dt;1). The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time. We provide the general solution of the renormalization group flows in terms of free fields, via Backlund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches...
Just-non-Lie nilpotent varieties of associative algebras
Finogenova, Olga
2011-01-01
We consider associative algebras over a field. An algebra variety is said to be {\\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$ is defined inductively by $[x_1, x_2, ..., x_n]=[[x_1, x_2, ..., x_{n-1}],x_n]$. It easy to see that every non-Lie nilpotent variety contains a minimal such subvariety. In the case of characteristic zero a complete description of the minimal non-Lie nilpotent (i.e. {\\em just-non-lie nilpotent}) varieties is found by Yu.Mal'cev. In the case of positive characteristic we reduce the problem of a description of such varieties to the case of {\\em prime} varieties.
2002-01-01
Some cohomology classes associated with an ideal in a Lie algebra, a Poisson structure on the basic functions algebra of contact structure, its Poisson cohomology and geometric (pre)quantization are considered from the algebraic point of view.
Giunashvili, Zakaria
2002-01-01
Some cohomology classes associated with an ideal in a Lie algebra, a Poisson structure on the basic functions algebra of contact structure, its Poisson cohomology and geometric (pre)quantization are considered from the algebraic point of view.
Harvey, J A; Harvey, Jeffrey A.; Moore, Gregory
1998-01-01
We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko \\& Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any CY 3-fold there are two canonically associated algebras exchanged by mirror symmetry.
Harvey, Jeffrey A.; Moore, Gregory
We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi-Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry.
Generalized NLS Hierarchies from Rational $W$ Algebras
Toppan, F
1994-01-01
Finite rational $\\cw$ algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of equations, by showing how to associate to a rational $\\cw$ algebra its corresponding hierarchy. We work out two examples: the $sl(2)/U(1)$ coset, leading to the Non-Linear Schr\\"{o}dinger hierarchy, and the $U(1)$ coset of the Polyakov-Bershadsky $\\cw$ algebra, leading to a $3$-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies.
On Lie Algebras in the Category of Yetter-Drinfeld Modules
Pareigis, B
1996-01-01
The category of Yetter-Drinfeld modules over a Hopf algebra (with bijektive antipode over a field) is a braided monoidal category. Given a Hopf algebra in this category then the primitive elements of this Hopf algebra do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in the category of Yetter-Drinfeld modules such that the set of primitive elements of a Hopf algebra is a Lie algebra in this sense. It has n-ary partially defined Lie multiplications on certain symmetric submodules of n- fold tensor products. They satisfy antisymmetry and Jacobi identities. Also the Yetter-Drinfeld module of derivations of an associative algebra in the category of Yetter- Drinfeld modules is a Lie algebra. Furthermore for each Lie algebra in the category of Yetter-Drinfeld modules there is a universal enveloping algebra which turns out to be a (braided) Hopf algebra in this category.
Characterization of Lie Higher Derivations on Triangular Algebras
Institute of Scientific and Technical Information of China (English)
Xiao Fei QI
2013-01-01
Let A and B be unital rings,and M be an (A,B)-bimodule,which is faithful as a left A-module and also as a right B-module.Let U =Tri(A,M,B) be the triangular algebra.In this paper,we give some different characterizations of Lie higher derivations on U.
Fixed Points of -Endomorphisms of a Free Metabelian Lie Algebra
Indian Academy of Sciences (India)
Naime Ekici; Demet Parlak Sönmez
2011-11-01
Let be a free metabelian Lie algebra of finite rank at least 2. We show the existence of non-trivial fixed points of an -endomorphism of and give an algorithm detecting them. In particular, we prove that the fixed point subalgebra Fix of an -endomorphism of is not finitely generated.
Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras
Directory of Open Access Journals (Sweden)
Luigi Accardi
2009-05-01
Full Text Available The present paper reviews some intriguing connections which link together a new renormalization technique, the theory of *-representations of infinite dimensional *-Lie algebras, quantum probability, white noise and stochastic calculus and the theory of classical and quantum infinitely divisible processes.
Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics
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Alex J. Dragt; Filippo Neri; Govindan Rangarajan; David Douglas; Liam M. Healy; Robert D. Ryne
1988-12-01
The purpose of this paper is to present a summary of new methods, employing Lie algebraic tools, for characterizing beam dynamics in charged-particle optical systems. These methods are applicable to accelerator design, charged-particle beam transport, electron microscopes, and also light optics. The new methods represent the action of each separate element of a compound optical system, including all departures from paraxial optics, by a certain operator. The operators for the various elements can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. This paper deals mostly with accelerator design and charged-particle beam transport. The application of Lie algebraic methods to light optics and electron microscopes is described elsewhere (1, see also 44). To keep its scope within reasonable bounds, they restrict their treatment of accelerator design and charged-particle beam transport primarily to the use of Lie algebraic methods for the description of particle orbits in terms of transfer maps. There are other Lie algebraic or related approaches to accelerator problems that the reader may find of interest (2). For a general discussion of linear and nonlinear problems in accelerator physics see (3).
Continual Lie algebras and noncommutative counterparts of exactly solvable models
Zuevsky, A.
2004-01-01
Noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of Saveliev-Vershik continual Lie algebras. Examples of noncommutative Liouville and sin/h-Gordon equations are given. The simplest soliton solution to the noncommutative sine-Gordon equation is found.
An Algorithm for the Decomposition of Semisimple Lie Algebras
Graaf, W.A. de
2001-01-01
We consider the problem of decomposing a semisimple Lie algebra dened over a eld of characteristic zero as a direct sum of its simple ideals The method is based on the decomposition of the action of a Cartan subalgebra An implementation of the algorithm in the system ELIAS is discussed at the end of
Leibniz Central Extension on a Block Lie Algebra
Institute of Scientific and Technical Information of China (English)
Qing Wang; Shaobin Tan
2007-01-01
Let B be the Lie algebra over C with basis {Lm,n | m, n ∈ Z, n≥0} and relations [Lm,n,Lm1 ,n1 ]=((n+1)m1-(n1+1)m) Lm+m1,n+n1. In this paper, we determine the second cohomology group and the second Leibniz cohomology group of B.
Non-solvable contractions of semisimple Lie algebras in low dimension
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Campoamor-Stursberg, R [Dpto. GeometrIa y TopologIa, Fac. CC. Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, E-28040 Madrid (Spain)
2007-05-18
The problem of non-solvable contractions of Lie algebras is analysed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras of dimension n {<=} 8, and obtain the non-solvable contractions of the latter class of algebras.
Classical affine W-algebras associated to Lie superalgebras
Energy Technology Data Exchange (ETDEWEB)
Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr [Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul 151-747 (Korea, Republic of)
2016-02-15
In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.
Adjoint representation of the graded Lie algebra osp(2/1; C) and its exponentiation
Ilyenko, K
2003-01-01
We construct explicitly the grade star Hermitian adjoint representation of osp(2/1; C) graded Lie algebra. Its proper Lie subalgebra, the even part of the graded Lie algebra osp(2/1; C), is given by su(2) compact Lie algebra. The Baker-Campbell-Hausdorff formula is considered and reality conditions for the Grassman-odd transformation parameters, which multiply the pair of odd generators of the graded Lie algebra, are clarified.
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, R [Dpto. GeometrIa y TopologIa, Fac. CC. Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, E-28040 Madrid (Spain); Low, S G [Austin, TX (United States)], E-mail: rutwig@mat.ucm.es, E-mail: Stephen.Low@alumni.utexas.net
2009-02-13
Given a semidirect product g=s oplus{sup {yields}} r of semisimple Lie algebras s and solvable algebras r, we construct polynomial operators in the enveloping algebra U(g) of g that commute with r and transform like the generators of s, up to a functional factor that turns out to be a Casimir operator of r. Such operators are said to generate a virtual copy of s in U(g), and allow us to compute the Casimir operators of g in a closed form, using the classical formulae for the invariants of s. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.
Extension of a quantized enveloping algebra by a Hopf algebra
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.
Recursion relations and branching rules for simple Lie algebras
Lyakhovsky, V D
1995-01-01
The branching rules between simple Lie algebras and its regular (maximal) simple subalgebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights \\Gamma specific for each injection. The structure of \\Gamma is easily deduced from the correspondence between the root systems of algebra and subalgebra. The recursion relations thus obtained give rise to simple and effective algorithm for branching rules. The details are exposed by performing the explicit decomposition procedure for A_{3} \\oplus u(1) \\rightarrow B_{4} injection.
Essays in the history of Lie groups and algebraic groups
Borel, Armand
2001-01-01
Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and con...
Valued Graphs and the Representation Theory of Lie Algebras
Directory of Open Access Journals (Sweden)
Joel Lemay
2012-07-01
Full Text Available Quivers (directed graphs, species (a generalization of quivers and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field. Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.
The Killing form and maximal toral subalgebra of the complete Lie algebra
Institute of Scientific and Technical Information of China (English)
孟道骥; 王淑苹
1996-01-01
Although the Killing form of a complete Lie algebra is degenerate in general,its restrictions to maximal toral subalgebras are still nondegenerate.This fact presents a criterion to simple complete Lie algebras in terms of root system.
A unified study of orthogonal polynomials via Lie algebra
Pathan, M. A.; Agarwal, Ritu; Jain, Sonal
2017-02-01
In this paper, we discuss some operators defined on Lie algebras for the purpose of deriving properties of some special functions. The method developed in this paper can also be used to study some other special functions of mathematical physics. We have established a general theorem concerning eigenvectors for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Further, using this result, we have obtained differential recurrence relations and differential equations for the extended Jacobi polynomials and the Gegenbauer polynomials. Results of many researchers; see for example Radulescu (1991), Mandal (1991), Pathan and Khan (2003), Humi, and the references therein, follow as special cases of our results.
Coadjoint orbits of reductive type of seaweed Lie algebras
Moreau, Anne
2011-01-01
A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), where q=Lie(Q). Due to results of M. Duflo, coadjoint representation of a quasi-reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as all seaweed subalgebras of gl(n) and describe the classes of their maximal reductive stabilisers.
A solvability criterion for the Lie algebra of derivations of a fat point
Schulze, Mathias
2009-01-01
We consider the Lie algebra of derivations of a zero dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result was motivated by and generalizes the solvability of the Yau algebra of an isolated hypersurface singularity.
Elementary n-Lie Algebras%基本n-Lie代数
Institute of Scientific and Technical Information of China (English)
白瑞蒲; 张艳艳
2007-01-01
In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras. We also give the relations between elementary n-algebras and E-algebras.
AUTOMRPPHISM GROUP OF LIE ALGEBRA C(t)d/dt
Institute of Scientific and Technical Information of China (English)
DU HONG
2003-01-01
The Lie algebra of derivations of rational function field C(t) is C(t)d/dt. The automorphism group of C(t) is well known as to be isomorphic to the projective linear group PGL(2, C). In this short note we prove that every automorphism of C(t)d/dt can be induced in a natural way from an automorphism of C(t).
Classical mechanics on noncommutative space with Lie-algebraic structure
Miao, Yan-Gang; Wang, Xu-Dong; Yu, Shao-Jie
2011-08-01
We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable tx˙-,(xx)˙-, and (xx)¨-dependence besides with the usual t-, x-, and x˙-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.
BRST Cohomology in Quantum Affine Algebra $U_q(\\widehat{sl_2})$
Konno, H
1994-01-01
Using free field representation of quantum affine algebra $U_q(\\widehat{sl_2})$, we investigate the structure of the Fock modules over $U_q(\\widehat{sl_2})$. The analisys is based on a $q$-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra $\\widehat {sl_2}$. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case ($q=1$), we obtain the irreducible highest weight representation space as a nontrivial cohomology group. This enables us to calculate a trace of the $q$-vertex operators over this space.
Analysis on singular spaces: Lie manifolds and operator algebras
Nistor, Victor
2016-07-01
We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called "Lie manifolds" -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.
POLYNOMIAL REPRESENTATIONS OF THE AFFINE NAPPI-WITTEN LIE ALGEBRA (H)4
Institute of Scientific and Technical Information of China (English)
Chen Xue; Huang Zhili
2012-01-01
In this paper,the representation theory for the affine Lie algebra （H）4 associated to the Nappi-Witten Lie algebra H4 is studied.Polynomial representations of the affine Nappi-Witten Lie algebra (H)4 are given.
Noncommutative physics on Lie algebras, Z_2^n lattices and Clifford algebras
Majid, S
2004-01-01
We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, $\\theta$-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice $(Z_2)^n$ and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on $Z_2\\times Z_2$ in a path integral approach.
Lie color 代数的商代数%Algebras of quotients of Lie color algebras
Institute of Scientific and Technical Information of China (English)
裴凤; 周建华
2004-01-01
介绍了Lie color 代数的一些性质,如素性、半素性、非退化性等.给出了Lie color 代数的商代数以及弱商代数的概念,并把Lie color 代数的素性和半素性推广到它的商代数上.利用没有非零零化子的理想对Lie color 代数的商代数进行刻画,证明了:若L是Lie color 代数Q的子代数,则Q是L的商代数当且仅当Q理想吸收于L.通过具体构造证明了每一个半素Lie color 代数都有极大商代数,并给出这个极大商代数的等价刻画.
Co-splitting of Simple Lie Algebras of Typ e A, D, E
Institute of Scientific and Technical Information of China (English)
Zhao Yu-e; Du Xian-kun
2015-01-01
In this paper, through a meticulous description of finite root system, a concrete comultiplication with an explicit action on the basis elements of finite dimensional simple Lie algebras of type A, D, E is constructed. Then any finite dimensional simple Lie algebra of type A, D, E is endowed with a new generalized Lie coalgebra splitting. This construction verifies the known existence of a co-split Lie structure on any finite dimensional complex simple Lie algebra.
Lie algebra lattices and strings on T-folds
Satoh, Yuji; Sugawara, Yuji
2017-02-01
We study the world-sheet conformal field theories for T-folds systematically based on the Lie algebra lattices representing the momenta of strings. The fixed point condition required for the T-duality twist restricts the possible Lie algebras. When the T-duality acts as a simple chiral reflection, one is left with the four cases, A 1 , D 2 r , E 7 , E 8, among the simple simply-laced algebras. From the corresponding Englert-Neveu lattices, we construct the modular invariant partition functions for the T-fold CFTs in bosonic string theory. Similar construction is possible also by using Euclidean even self-dual lattices. We then apply our formulation to the T-folds in the E 8 × E 8 heterotic string theory. Incorporating non-trivial phases for the T-duality twist, we obtain, as simple examples, a class of modular invariant partition functions parametrized by three integers. Our construction includes the cases which are not reduced to the free fermion construction.
Lie algebra lattices and strings on T-folds
Satoh, Yuji
2016-01-01
We study the world-sheet conformal field theories for T-folds systematically based on the Lie algebra lattices representing the momenta of strings. The fixed point condition required for the T-duality twist restricts the possible Lie algebras. When the T-duality acts as a simple chiral reflection, one is left with the four cases, $A_1, D_{2r}, E_7, E_8$, among the simple simply-laced algebras. From the corresponding Englert-Neveu lattices, we construct the modular invariant partition functions for the T-fold CFTs in bosonic string theory. Similar construction is possible also by using Euclidean even self-dual lattices. We then apply our formulation to the T-folds in the $E_8 \\times E_8$ heterotic string theory. Incorporating non-trivial phases for the T-duality twist, we obtain, as simple examples, a class of modular invariant partition functions parametrized by three integers. Our construction includes the cases which are not reduced to the free fermion construction.
Helminck, G.F.; Helminck, A.G.; Panasenko, E.A.
2013-01-01
We split the algebra of pseudodifferential operators in two different ways into the direct sum of two Lie subalgebras and deform the set of commuting elements in one subalgebra in the direction of the other component. The evolution of these deformed elements leads to two compatible systems of Lax eq
Some Remarks Concerning the Invariants of Rank One Solvable Real Lie Algebras
Institute of Scientific and Technical Information of China (English)
Rutwig Campoamor-Stursberg
2005-01-01
A corrected and completed list of six dimensional real Lie algebras with five dimensional nilradical is presented. Their invariants for the coadjoint representation are computed and some results on the invariants of solvable Lie algebras in arbitrary dimension whose nilradical has codimension one are also given. Specifically, it is shown that any rank one solvable Lie algebra of dimension n without invariants determines a family of (n +2k)-dimensional algebras with the same property.
Surfaces immersed in Lie algebras associated with elliptic integrals
Energy Technology Data Exchange (ETDEWEB)
Grundland, A M; Post, S, E-mail: grundlan@crm.umontreal.ca, E-mail: post@crm.umontreal.ca [Centre de Recherches Mathematiques, Universite de Montreal, Montreal CP6128, QC H3C 3J7 (Canada)
2012-01-13
The objective of this work is to adapt the Fokas-Gel'fand immersion formula to ordinary differential equations written in the Lax representation. The formalism of generalized vector fields and their prolongation structure is employed to establish necessary and sufficient conditions for the existence and integration of immersion functions for surfaces in Lie algebras. As an example, a class of second-order, integrable, ordinary differential equations is considered and the most general solutions for the wavefunctions of the linear spectral problem are found. Several explicit examples of surfaces associated with Jacobian and P-Weierstrass elliptic functions are presented. (paper)
ENDOMORPHISMS OF LIE ALGEBRA F[t]d/dt
Institute of Scientific and Technical Information of China (English)
DU Hong
2004-01-01
Let F be a field of characteristic zeroWn =F[t +1/1,t +1/2,…,t +1/n]а/аt1+ are simple infinite dimensional Lie algebraIn Zhao's paper, it was conjectured thatEnd(W,n+) - {0} = Aut(Wn+) and it was proved that the validity of this conjecture im-plies the validity of the well-known Jacobian conjectureIn this short note, we check theconjecture above for n = 1We show End(W+1) - {0} = Aut(W1+).
Generalized Lotka—Volterra systems connected with simple Lie algebras
Charalambides, Stelios A.; Damianou, Pantelis A.; Evripidou, Charalambos A.
2015-06-01
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.
GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS
Institute of Scientific and Technical Information of China (English)
陈正新
2014-01-01
Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) 6= 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.
International Workshop "Groups, Rings, Lie and Hopf Algebras"
2003-01-01
The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.
An introduction to some novel applications of Lie algebra cohomology and physics
de Azcárraga, J A; Bueno, J C P
1998-01-01
After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5. Higher order simple and SH Lie algebras 6. Higher order generalized Poisson structures 7. Relative cohomology, coset spaces and effective WZW actions
A Poincaré-Birkhoff-Witt Theorem for Generalized Color Lie Algebras
Bautista, C
1997-01-01
A proof of Poincaré-Birkhoff-Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich $S$-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type $A_{n}$ and $M_{p,q,\\epsilon}(n,K)$, which is a non-standard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the Poincaré-Birkhoff-Witt theorem and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras.
The relativistic invariant Lie algebra for the kinematical observables in quantum space-time
Khrushchov, V V
2003-01-01
The deformation of the canonical algebra for the kinematical observables in Minkowski space has been considered under the condition of Lorentz invariance. A new relativistic invariant algebra depends on the fundamental constants $M$, $L$ and $H$ with the dimensionality of mass, length and action, respectively. In some limit cases the algebra obtained goes over into the well-known Snyder or Yang algebras. In general case the algebra represents a class of Lie algebras, which are either simple algebras, or semidirect sums of simple algebras integrable ones. T and C noninvariance for certain algebras of this class have been elucidated.
Irreducible Highest Weight Representations Of The Simple n-Lie Algebra
Balibanu, Dana
2010-01-01
A. Dzhumadil'daev classified all irreducible finite dimensional representations of the simple n-Lie algebra. Using a slightly different approach, we obtain in this paper a complete classification of all irreducible, highest weight modules, including the infinite-dimensional ones. As a corollary we find all primitive ideals of the universal enveloping algebra of this simple n-Lie algebra.
Some Results on Metric n-Lie Algebras
Institute of Scientific and Technical Information of China (English)
Rui Pu BAI; Wan Qing WU; Zhen Heng LI
2012-01-01
We study the structure of a metric n-Lie algebra G over the complex field C.Let (G) =S(+)R be the Levi decomposition,where R is the radical of (G) and S is a strong semisimple subalgebra of (G).Denote by m((G)) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R⊥the orthogonal complement of R.We obtain the following results.As S-modules,R⊥ is isomorphic to the dual module of (G)/R.The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on (G) is equal to that of the vector space of certain linear transformations on (G); this dimension is greater than or equal to m((G)) + 1.The centralizer of R in (G) is equal to the sum of all minimal ideals; it is the direct sum of R⊥ and the center of (G).Finally,(G) has no strong semisimple ideals if and only if R⊥ (C) R.
Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras
Boykin, P O; Tiep, P H; Wocjan, P; Sitharam, Meera; Tiep, Pham Huu; Wocjan, Pawel
2005-01-01
We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that...
The Realization of 4-dimensional 3-Lie Algebras%4维3-Lie代数的实现
Institute of Scientific and Technical Information of China (English)
刘建波; 张艳艳; 张知学
2007-01-01
In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.
Field Theories on Canonical and Lie-Algebra Noncommutative Spacetimes
Amelino-Camelia, G; Doplicher, L; Amelino-Camelia, Giovanni; Arzano, Michele; Doplicher, Luisa
2002-01-01
Field theories on canonical noncommutative spacetimes, which are being studied also in connection with string theory, and on $\\kappa$-Minkowski spacetime, which is a popular example of Lie-algebra noncommutative spacetime, can be naturally constructed by introducing a suitable generating functional for Green functions in energy-momentum space. Direct reference to a star product is not necessary. It is sufficient to make use of the simple properties that the Fourier transform preserves in these spacetimes and establish the rules for products of wave exponentials that are dictated by the non-commutativity of the coordinates. The approach also provides an elementary description of "planar" and "non-planar" Feynman diagrams. We also comment on the rich phenomenology emerging from the analysis of these theories.
Field Theories on Canonical and Lie-Algebra Noncommutative Spacetimes
Amelino-Camelia, G.; Arzano, M.; Doplicher, L.
2003-01-01
Field theories on canonical noncommutative spacetimes, which are being studied also in connection with string theory, and on k-Minkowski spacetime, which is a popular example of Lie-algebra noncommutative spacetime, can be naturally constructed by introducing a suitable generating functional for Green functions in energy-momentum space. Direct reference to a star product is not necessary. It is sufficient to make use of the simple properties that the Fourier transform preserves in these spacetimes and establish the rules for products of wave exponentials that are dictated by the non-commutativity of the coordinates. The approach also provides an elementary description of "planar" and "non-planar" Feynman diagrams. We also comment on the rich phenomenology emerging from the analysis of these theories.
Lie algebraic similarity transformed Hamiltonians for lattice model systems
Wahlen-Strothman, Jacob M.; Jiménez-Hoyos, Carlos A.; Henderson, Thomas M.; Scuseria, Gustavo E.
2015-01-01
We present a class of Lie algebraic similarity transformations generated by exponentials of two-body on-site Hermitian operators whose Hausdorff series can be summed exactly without truncation. The correlators are defined over the entire lattice and include the Gutzwiller factor ni ↑ni ↓ , and two-site products of density (ni ↑+ni ↓) and spin (ni ↑-ni ↓) operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with polynomial computational cost. The proposed similarity transformation generates locally weighted orbital transformations of the reference determinant. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one- and two-dimensional repulsive Hubbard model where it yields accurate results for small and medium sized interaction strengths.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
Yu, Zhang; Zhang, Yufeng
2009-01-15
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.
A Realization of Hom-Lie Algebras by Iso-Deformed Commutator Bracket
Xiuxian Li
2013-01-01
We construct classical Iso-Lie and Iso-Hom-Lie algebras in $gl(V)$ by twisted commutator bracket through Iso-deformation. We prove that they are simple. Their Iso-automorphisms and isotopies are also presented.
New Applications of a Kind of Infinitesimal-Operator Lie Algebra
Directory of Open Access Journals (Sweden)
Honwah Tam
2016-01-01
Full Text Available Applying some reduced Lie algebras of Lie symmetry operators of a Lie transformation group, we obtain an invariant of a second-order differential equation which can be generated by a Euler-Lagrange formulism. A corresponding discrete equation approximating it is given as well. Finally, we make use of the Lie algebras to generate some new integrable systems including (1+1 and (2+1 dimensions.
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, Rutwig [Depto. Geometria y Topologia, Fac. cc. Matematicas UCM, E-28040 Madrid (Spain)
2004-10-08
We show that the Casimir operators of the semidirect products G{sub 2} {rvec {circle_plus}}{sub 2{gamma}{sub (a,b){circle_plus}{Lambda}{sub (0,0)}}}h of the exceptional Lie algebra G{sub 2} and a Heisenberg algebra h can be constructed explicitly from the Casimir operators of G{sub 2}.
The centralizer of an element in a Lie algebra of type L
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
［1］Osbom, J. M., Zhao, K., Infinite dimensional Lie algebras of type L, Comm. Alg., 2003, 31(5): 2445-2470.［2］Osbom, J. M., Zhao, K., Generalized cartan type K Lie algebras in characteristic 0, Comm. Alg., 1997, 25:3325-3360.［3］Osborn, J. M., Zhao, K., Infinite dimensional Lie algebras of generalized Block type, Proc. of AMS, 1999,127(6): 1641-1650.［4］Djokovic, D. Z., Zhao, K., Derivations, isomorphisms,and second cohomology of generalized witt algebras,Trans. Amer. Math. Soc., 1998, 350(2): 643-464.［5］Djokovic, D. Z., Zhao, K., Derivations, isomorphisms, and second cohomology of Block algebras, Algebra Colloquium, 1996, 3(3): 245-272.［6］Djokovic, D. Z., Zhao, K., Some infinite dimensional simple Lie algebras related to those of Block, J. Pure and Applied Alg., 1998, 127(2): 153-165.［7］Djokovic, D. Z., Zhao, K., Derivations, generalized cartan type S Lie algebras of charateristic 0, J. Alg., 1997,193:144-179.［8］Osborn, J. M., New simple infinite-dimensional Lie algebras of characteristic 0, J. Alg., 1996, 185: 820-835.［9］Osborn, J. M., Derivations and isomorphisms of Lie algebras of characteristic of 0, Studies in Advanced Math.,1997, 49(1): 95-108.［10］Osborn, J. M., Automorphisms of the Lie algebras W* in characteristic 0, Canadian J. Math., 1997, 49(1):119-132.［11］Osborn, J. M., Passman, D. S., Derivations of skew polynomial rings, J. Alg., 1995, 176:417-448.［12］Rudakov, A. N., Groups of automorphisms of infinite dimensional simple Lie algebras, Izv. Nauk SSSR, Ser.Mat. Tom, 1969, 33(4): 707-722.
On Complete Lie Algebras and Lie Groups%关于完备李群与完备李代数
Institute of Scientific and Technical Information of China (English)
梁科; 邓少强
2001-01-01
孟道骥等对完备李代数作了系统的研究并已获得很多基本和重要的结果.本文给出完备李群与完备李代数的某些关系.%Daoji Meng and others have made a systematic study on complete Lie algebras and obtained some basic and important conclusions. In this paper, we will investigate relations between complete Lie groups and complete Lie algebras.
Theoretical study of nonlinear triatomic molecular potential energy surfaces:Lie algebraic method
Institute of Scientific and Technical Information of China (English)
郑雨军; 丁世良
2000-01-01
Triatomic molecular potential energy surfaces (PES) are obtained by using coherent state to take the classical limits of algebraic Hamiltonian. The algebraic Hamiltonian for bent tria-tomic molecules can be obtained using Lie algebraic method (the expansion coefficients are obtained by fitting spectroscopic data). This PES is applied to H2O molecule, and good results are obtained.
Exposition on affine and elliptic root systems and elliptic Lie algebras
Azam, Saeid; Yousofzadeh, Malihe
2009-01-01
This is an exposition in order to give an explicit way to understand (1) a non-topological proof for an existence of a base of an affine root system, (2) a Serre-type definition of an elliptic Lie algebra with rank =>2, and (3) the isotropic root multiplicities of those elliptic Lie algebras.
Test elements of direct sums and free products of free Lie algebras
Indian Academy of Sciences (India)
Cennet Eskal; Naime Ekici
2016-02-01
We give a characterization of test elements of a direct sum of free Lie algebras in terms of test elements of the factors. In addition, we construct certain types of test elements and we prove that in a free product of free Lie algebras, product of the homogeneous test elements of the factors is also a test element.
Enhanced gauge groups in N=4 topological amplitudes and Lorentzian Borcherds algebras
Hohenegger, Stefan; Persson, Daniel
2011-11-01
We continue our study of algebraic properties of N=4 topological amplitudes in heterotic string theory compactified on T2, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group h⊂e8⊕e8, i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra g++. The latter is obtained through double extension of the complement g=(e8⊕e8)/h. The infinite product is automorphic with respect to a finite index subgroup of the full T-duality group SO(2,18;Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra G(g++), which is an ’automorphic correction’ of g++. We explicitly give the root multiplicities of G(g++) for a number of examples.
Enhanced Gauge Groups in N=4 Topological Amplitudes and Lorentzian Borcherds Algebras
Hohenegger, Stefan
2011-01-01
We continue our study of algebraic properties of N=4 topological amplitudes in heterotic string theory compactified on T^2, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group h \\subset e_8 + e_8, i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra g^{++}. The latter is obtained through double extension of the complement g= (e_8 + e_8)/h. The infinite product is automorphic with respect to a finite index subgroup of the full T-duality group SO(2,18;Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra G(g^{++}), which is an 'automorphic correction' of g^{++}. We explicitly give the root multiplicities of G(g^{++}) for a number of examples.
Screenings and a universal Lie-de Rham cocycle
Ginzburg, V; Ginzburg, Victor; Schechtman, Vadim
1997-01-01
Feigin and Fuchs have given a well-known construction of intertwining operators between "Fock-type" modules over the Virasoro algebra. The intertwiners are obtained via contour integration of certain "screening operators" over top homology classes of a configuration space. The main observation of the present paper is that the screening operators contain more information. Specifically, at the chain level, the screening operators provide a certain canonical cocycle of the Virasoro (resp. affine Kac-Moody) algebra with coefficients in the de Rham complex of an operator-valued local system on the configuration space. This way we obtain canonical morphisms from higher homology groups of the above local systems to appropriate higher $\\Ext$-groups between the Fock space representations. Our construction is motivated by, and in a special case reduces to the construction of Bowknegt et al, see [BMP1], [BMP2].
Weight Representations of Admissible Affine Vertex Algebras
Arakawa, Tomoyuki; Futorny, Vyacheslav; Ramirez, Luis Enrique
2017-08-01
For an admissible affine vertex algebra {V_k{(\\mathfrak{g})}} of type A, we describe a new family of relaxed highest weight representations of {V_k{(\\mathfrak{g})}}. They are simple quotients of representations of the affine Kac-Moody algebra {\\widehat{\\mathfrak{g}}} induced from the following {\\mathfrak{g}}-modules: (1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from {\\mathfrak{sl}_2}; (2) all Gelfand-Tsetlin modules in the principal nilpotent orbit that are induced from {\\mathfrak{sl}_3}; (3) all simple Gelfand-Tsetlin modules over {\\mathfrak{sl}_3}. This in particular gives the classification of all simple positive energy weight representations of {V_k{(\\mathfrak{g})}} with finite dimensional weight spaces for {\\mathfrak{g}=\\mathfrak{sl}_3}.
Categorification of skew-symmetrizable cluster algebras
Demonet, Laurent
2009-01-01
We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with the action of a finite group G, we construct a G-equivariant mutation on the set of maximal rigid G-invariant objects of C. Using an appropriate cluster character, we can then attach to these data an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac-Moody groups, generalizing to the non simply-laced case several results of Gei\\ss-Leclerc-Schr\\"oer.
Common aspects of q-deformed Lie algebras and fractional calculus
Herrmann, Richard
2010-01-01
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. A new class of fractional q-deformed Lie algebras is proposed, which for the first time allows a smooth transition between different Lie algebras. For the fractional harmonic oscillator, the corresponding fractional q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool to describe e.g. the ground state spectra of even-even nuclei. In addition, the equivalence of rotational and vibrational spectra for fractional q-deformed Lie algebras is shown and the $B_\\alpha(E2)$ values for the fractional q-deformed symmetric rotor are calculated. A first interpretation of half integer representations of the fractional rotation group is given in terms of a description of $K=1/2^-$ band spectra of odd-even nuclei.
On the Lie Symmetry Algebras of the Stationary Schrödinger and Pauli Equations
Boldyreva, M. N.; Magazev, A. A.
2017-02-01
A general method for constructing first-order symmetry operators for the stationary Schrödinger and Pauli equations is proposed. It is proven that the Lie algebra of these symmetry operators is a one-dimensional extension of some subalgebra of an e(3) algebra. We also assemble a classification of stationary electromagnetic fields for which the Schrödinger (or Pauli) equation admits a Lie algebra of first-order symmetry operators.
Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach
Panahi, H.; Baradaran, M.; Savadi, A.
2015-10-01
Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.
Linear Commuting Maps on Parab olic Subalgebras of Finite-dimensional Simple Lie Algebras
Institute of Scientific and Technical Information of China (English)
CHEN Zheng-xin; WANG Bing
2014-01-01
A map ϕ on a Lie algebra g is called to be commuting if [ϕ(x), x] = 0 for all x∈g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapϕon P is commuting if and only ifϕis a scalar multiplication map on P .
Some quantum Lie algebras of type D{sub n} positive
Energy Technology Data Exchange (ETDEWEB)
Bautista, Cesar [Facultad de Ciencias de la Computacion, Benemerita Universidad Autonoma de Puebla, Edif 135, 14 sur y Av San Claudio, Ciudad Universitaria, Puebla Pue. CP 72570 (Mexico); Juarez-Ramirez, Maria Araceli [Facultad de Ciencias Fisico-Matematicas, Benemerita Universidad Autonoma de Puebla, Edif 158 Av San Claudio y Rio Verde sn Ciudad Universitaria, Puebla Pue. CP 72570 (Mexico)
2003-03-07
A quantum Lie algebra is constructed within the positive part of the Drinfeld-Jimbo quantum group of type D{sub n}. Our quantum Lie algebra structure includes a generalized antisymmetry property and a generalized Jacobi identity closely related to the braid equation. A generalized universal enveloping algebra of our quantum Lie algebra of type D{sub n} positive is proved to be the Drinfeld-Jimbo quantum group of the same type. The existence of such a generalized Lie algebra is reduced to an integer programming problem. Moreover, when the integer programming problem is feasible we show, by means of the generalized Jacobi identity, that the Poincare-Birkhoff-Witt theorem (basis) is still true.
Anti-commutative Gr(o)bner-Shirshov basis of a free Lie algebra
Institute of Scientific and Technical Information of China (English)
BOKUT L.A.; CHEN YuQun; LI Yu
2009-01-01
The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grobner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.
Anti-commutative Grbner-Shirshov basis of a free Lie algebra
Institute of Scientific and Technical Information of China (English)
BOKUT; L.; A.
2009-01-01
The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.
Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups
Breev, A. I.; Mosman, E. A.
2016-12-01
The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.
Institute of Scientific and Technical Information of China (English)
冯东太; 丁世良; 王美山
2003-01-01
The highly excited vibrational states of asymmetric linear tetratomic molecules are studied in the framework of Lie algebra. By using symmetric group U1(4) U2(4) U3(4), we construct the Hamiltonian that includes not only Casimir operators but also Majorana operators M12,M13 and M23, which are useful for getting potential energy surface and force constants in Lie algebra method. By Lie algebra treatment, we obtain the eigenvalues of the Hamiltonian, and make the concrete calculation for molecule C2HF.
A new approach to tolerance analysis method based onthe screw and the Lie Algebra of Lie Group
Zhai, X. C.; Du, Q. G.; Wang, W. X.; Wen, Q.; Liu, B. S.; Sun, Z. Q.
2016-11-01
Tolerance analysis refers to the process of establishing mapping relations between tolerance features and the target feature along the dimension chain. Traditional models for tolerance analysis are all based on rigid body kinematics, and they adopt the Homogeneous Transformation Matrix to describe feature variation and accumulation. However, those models can hardly reveal the nature of feature variations. This paper proposes a new tolerance analysis method based on the screw and the Lie Algebra of Lie Group, which describes feature variation as the screw motion, and completely maps the twist, an element of the Lie Algebra, to the Lie Group that represents the feature configuration space. Thus, the analysis can be conducted in a more succinct and direct way. In the end, the method is applied in an example and proven to be robust and effective.
Lie $3-$algebra and super-affinization of split-octonions
Giardino, Sergio; Carrion, Hector L.
2010-01-01
The purpose of this study is to extend the concept of a generalized Lie $3-$ algebra, known to the divisional algebra of the octonions $\\mathbb{O}$, to split-octonions $\\mathbb{SO}$, which is non-divisional. This is achieved through the unification of the product of both of the algebras in a single operation. Accordingly, a notational device is introduced to unify the product of both algebras. We verify that $\\mathbb{SO}$ is a Malcev algebra and we recalculate known relations for the structur...
Subcategories of fixed points of mutations in root categories with type n-1
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type n-2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type n-1.
Algebras of Complete Hörmander Vector Fields, and Lie-Group Construction
Directory of Open Access Journals (Sweden)
Andrea Bonfiglioli
2014-12-01
Full Text Available The aim of this note is to characterize the Lie algebras g of the analytic vector fields in RN which coincide with the Lie algebras of the (analytic Lie groups defined on RN (with its usual differentiable structure. We show that such a characterization amounts to asking that: (i g is N-dimensional; (ii g admits a set of Lie generators which are complete vector fields; (iii g satisfies Hörmander’s rank condition. These conditions are necessary, sufficient and mutually independent. Our approach is constructive, in that for any such g we show how to construct a Lie group G = (RN, * whose Lie algebra is g. We do not make use of Lie’s Third Theorem, but we only exploit the Campbell-Baker-Hausdorff-Dynkin Theorem for ODE’s.
Boson permutation and parity operators: Lie algebra and applications
Energy Technology Data Exchange (ETDEWEB)
Campos, Richard A. [Department of Physics and Astronomy, Lehman College, City University of New York, 250 Bedford Boulevard West, Bronx, NY 10468-1589 (United States)]. E-mail: richard.campos@mailaps.org; Gerry, Christopher C. [Department of Physics and Astronomy, Lehman College, City University of New York, 250 Bedford Boulevard West, Bronx, NY 10468-1589 (United States)
2006-08-14
We show that dichotomic permutation and parity operators for a two-dimensional boson system form an su(2) algebra with a unitary operator that relates, in quantum optics, to a balanced beamsplitter. The algebra greatly simplifies the input-output transformations of states through quantum nonlinear systems such as the Kerr interferometer or the kicked top.
On Polynomial Representations of Lie Algebras%Lie代数的多项式表示
Institute of Scientific and Technical Information of China (English)
陈酌; 贺龙光; 钟德寿
2006-01-01
We study polynomial representations of finite dimensional (R or C) Lie algebras. As a total classification, we show that there are altogether three types of such nontrivial representations and give their subtle structures.
Institute of Scientific and Technical Information of China (English)
郑雨军; 丁世良
2000-01-01
The vibrational excitations of bent triatomic molecules are studied by using Lie algebra. The RMS error of fitting 30 spectroscopic data is 1.66 cm-1 for SO2. The results show that the expansion of a molecular algebraic Hamiltonian can well describe the experimental data. And the total vibrational levels can be calculated using this Hamiltonian. At the same time, the potential energy surface can also be obtained with the algebraic Hamiltonian.
Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth-Death Processes with Immigration
Ohkubo, Jun
2014-10-01
Analytical solutions for time-inhomogeneous linear birth-death processes with immigration are derived. While time-inhomogeneous linear birth-death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth-death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.
Upper Triangular Matrix of Lie Algebra and a New Discrete Integrable Coupling System
Institute of Scientific and Technical Information of China (English)
YU Fa-Jun; ZHANG Hong-Qing
2007-01-01
The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations.Correspondingly,a feasible way to construct integrable couplings is presented.A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy.It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.
Lie algebra solution of population models based on time-inhomogeneous Markov chains
House, Thomas
2011-01-01
Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical and social applications. This paper presents the Lie algebraic method, and applies it to three biologically well motivated examples. The result of this is a solution form that is often highly computationally advantageous.
Exact Solutions of Two Coupled Harmonic Oscillators Related to the Sp(4, R) Lie Algebra
Institute of Scientific and Technical Information of China (English)
PAN Feng; DAI LianRong
2001-01-01
Exact solutions of the eigenvalue problem of two coupled harmonic oscillators related to the Sp(4, R) Lie algebra are derived by using an algebraic method. It is found that the energy spectrum of the system is determined by one-boson excitation energies built on a vector coherent state of Sp(4, R) U(2).``
Fuzzy Torus and q-Deformed Lie Algebra
Nakayama, R
2006-01-01
It will be shown that the defining relations for fuzzy torus and deformed (squashed) sphere proposed by J. Arnlind, et al (hep-th/0602290) can be rewriten as a new algebra which contains q-deformed commutators. The quantum parameter q (|q|=1) is a function of \\hbar. It is shown that the q --> 1 limit of the algebra with the parameter \\mu <0 describes fuzzy S^2 and that the squashed S^2 with q \
Higher powers of analytical operators and associated ∗-Lie algebras
Ettaieb, Aymen; Khalifa, Narjess Turki; Ouerdiane, Habib; Rguigui, Hafedh
2016-06-01
We introduce a new product of two test functions denoted by f□g (where f and g in the Schwartz space 𝒮(ℝ)). Based on the space of entire functions with θ-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product f□g, such operators give us a new representation of the centerless Virasoro-Zamolodchikov-ω∞∗-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function f as any Hermite function. Replacing the classical pointwise product f ṡ g of two test functions f and g by f□g, we prove the existence of new ∗-Lie algebras as counterpart of the classical powers of white noise ∗-Lie algebra, the renormalized higher powers of white noise (RHPWN) ∗-Lie algebra and the second quantized centerless Virasoro-Zamolodchikov-ω∞∗-Lie algebra.
Solvability of a Lie algebra of vector fields implies their integrability by quadratures
Cariñena, J. F.; Falceto, F.; Grabowski, J.
2016-10-01
We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures.
Energy Technology Data Exchange (ETDEWEB)
Abedi-Fardad, J., E-mail: j.abedifardad@bonabu.ac.ir [Department of Mathematics, Bonab University, Tabriz (Iran, Islamic Republic of); Rezaei-Aghdam, A., E-mail: rezaei-a@azaruniv.edu [Department of Physics, Azarbaijan Shahid Madani University, 53714-161 Tabriz (Iran, Islamic Republic of); Haghighatdoost, Gh., E-mail: gorbanali@azaruniv.edu [Department of Mathematics, Bonab University, Tabriz (Iran, Islamic Republic of); Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz (Iran, Islamic Republic of)
2014-05-15
We construct integrable and superintegrable Hamiltonian systems using the realizations of four dimensional real Lie algebras as a symmetry of the system with the phase space R{sup 4} and R{sup 6}. Furthermore, we construct some integrable and superintegrable Hamiltonian systems for which the symmetry Lie group is also the phase space of the system.
Lorentzian Lie (3-)algebra and toroidal compactification of M/string theory
Ho, Pei-Ming; Shiba, Shotaro
2009-01-01
We construct a class of Lie 3-algebras with an arbitrary number of pairs of generators with Lorentzian signature metric. Some examples are given and corresponding BLG models are studied. We show that such a system in general describes a supersymmetric massive vector multiplets after the ghost fields are Higgsed. Simple systems with nontrivial interaction are realized by infinite dimensional Lie 3-algebras associated with the loop algebras. The massive fields are then naturally identified with the Kaluza-Klein modes by the toroidal compactification triggered by the ghost fields. For example, Dp-brane with an (infinite dimensional) affine Lie algebra symmetry $\\hat g$ can be identified with D(p+1)-brane with gauge symmetry $g$.
The Adapted Ordering Method for Lie Algebras and Superalgebras and their Generalizations
Gato-Rivera, Beatriz
2007-01-01
In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras in two dimensions. It allows: to determine maximal dimensions for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this article we present the Adapted Ordering Method for general Lie algebras and superalgebras, and their generalizations, provided they can be triangulated. We also review briefly the results obtained for the Virasoro algebra and for the N=2 and Ramond N=1 superconformal algebras.
Poisson-Lie T-Duality and Bianchi Type Algebras
Jafarizadeh, M A
1999-01-01
All Bianchi bialgebras have been obtained. By introducing a non-degenerate adjoint invariant inner product over these bialgebras the associated Drinfeld doubles have been constructed, then by calculating the coupling matrices for these bialgebras several \\sigma-models with Poisson-Lie symmetry have been obtained. Two simple examples as prototypes of Poisson-Lie dual models have been given.
Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability
Institute of Scientific and Technical Information of China (English)
CHEN ZHENG-XIN; CHEN QIONG
2012-01-01
Let Mn be the algebra of all n × n complex matrices and gl(n,C) be the general linear Lie algebra,where n ≥ 2.An invertible linear map ?:gl(n,C) →gl(n,C) preserves solvability in both directions if both ? and ?-1 map every solvable Lie subalgebra of gl(n,C) to some solvable Lie subalgebra.In this paper we classify the invertible linear maps preserving solvability on gl(n,C) in both directions.As a sequence,such maps coincide with the invertible linear maps preserving commutativity on Mn in both directions.
Characterizing ξ-Lie Multiplicative Isomorphisms on Von Neumann Algebras
Directory of Open Access Journals (Sweden)
Yamin Song
2014-01-01
Full Text Available Let ℳ and be von Neumann algebras without central summands of type I1. Assume that ξ∈ℂ with ξ≠1. In this paper, all maps Φ:ℳ→ satisfying ΦAB-ξBA=ΦAΦB-ξΦBΦ(A are characterized.
Gauge Theories on Open Lie Algebra Noncommutative Spaces
Agarwal, A.; Akant, L.
It is shown that noncommutative spaces, which are quotients of associative algebras by ideals generated by highly nonlinear relations of a particular type, admit extremely simple formulae for deformed or star products. Explicit construction of these star products is carried out. Quantum gauge theories are formulated on these spaces, and the Seiberg-Witten map is worked out in detail.
Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors
Açık, Özgür; Ertem, Ümit
2016-08-01
We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six- and ten-dimensional cases. For five- and eleven-dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.
The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials
Finkel, F; Finkel, Federico; Kamran, Niky
1996-01-01
We prove that the scalar and $2\\times 2$ matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general graphical method which does not require the modules to be irreducible under the action of the corresponding Lie (super)algebra. This method can be generalized to modules of polynomials in an arbitrary number of variables. We give generic examples of partially solvable differential operators which are not Lie algebraic. We show that certain vector-valued modules give rise to new realizations of finite-dimensional Lie superalgebras by first-order differential operators.
Dobrev, V K
2013-01-01
In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of {\\it parabolic relation} between two non-compact semisimple Lie algebras $\\cal G$ and $\\cal G'$ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra $E_{7(7)}$ which is parabolically related to the CLA $E_{7(-25)}$. Other interesting examples are the orthogonal algebras $so(p,q)$ all of which are parabolically related to the conformal algebra $so(n,2)$ with $p+q=n+2$, the parabolic subalgebras including the Lorentz subalgebra $so(n-1,1)$ and its analogs ...
Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
Meljanac, Stjepan; Krešić–Jurić, Saša; Martinić, Tea
2017-07-01
This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0, we construct a Lie superalgebra g =g0⊕g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g ) gives the commutation relations between monomials in U (g0 ) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0 ) =dim(g1 ) , we also find a realization of the exterior derivative on U (g0 ) . The realizations of these geometric objects yield a bicovariant differential calculus on U (g0 ) as a deformation of the standard calculus on the Euclidean space.
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations
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Rutwig Campoamor-Stursberg
2016-03-01
Full Text Available A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.
Institute of Scientific and Technical Information of China (English)
RUAN Dong; YUAN Jing; JIA Yu-Feng; SUN Hong-Zhou
2001-01-01
The explicit expressions for indecomposable representations ofnine square-root Lie algebras of vector type,Rレイ (v,ィ = 0,+-1),are obtained on the space of universal enveloping algebra of two-state Heisenberg-Weyl algebra,the invariant subspaces and the quotient spaces.From Fock representations corresponding to these indecomposablerepresentations,the inhomogeneous boson realizations of are given.The expectation values of R in the angularmomentum coherent states are calculated as well as the corresponding classical limits.
Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type
Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah
2015-11-01
We propose a noncommutative version of the Euclidean Lie algebra E 2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.
Varchenko, A N
1995-01-01
This book recounts the connections between multidimensional hypergeometric functions and representation theory. In 1984, physicists Knizhnik and Zamolodchikov discovered a fundamental differential equation describing correlation functions in conformal field theory. The equation is defined in terms of a Lie algebra. Kohno and Drinfeld found that the monodromy of the differential equation is described in terms of the quantum group associated with the Lie algebra. It turns out that this phenomenon is the tip of the iceberg. The Knizhnik-Zamolodchikov differential equation is solved in multidimens
Lie algebras for time evolution with applications from chaos studies to spintronics
Wendler, Tim G.; Berrondo, Manuel; Beus, Ty; Sayer, Ryan T.; van Huele, Jean-Francois S.
2012-10-01
We illustrate the power of Lie algebras in computing the time evolution of quantum systems with time-dependent physical parameters. By factorizing the quantum mechanical time evolution operator and using the linear independence of the Lie algebra generators, we reduce the operator equations to systems of coupled ordinary differential equations of scalar functions applicable to a variety of dynamical systems. We use the results to explore the possibility of detecting chaos in quantum nonlinear oscillators based on criteria from classical chaos studies and to follow spin currents in time-dependent spin-orbit coupled media.
Characterization of Lie Derivations on von Neumann Algebras
Qi, XIaofei
2012-01-01
Let ${\\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$ and $\\xi\\in{\\mathbb C}$ a scalar. It is shown that an additive map $L$ on $\\mathcal M$ satisfies $L(AB-\\xi BA)=L(A)B-\\xi BL(A)+L(B)A-\\xi AL(B)$ whenever $A,B\\in{\\mathcal M}$ with $AB=0$ if and only if one of the following statements holds: (1) $\\xi=1$, $L=\\varphi+f$, where $\\varphi$ is an additive derivation on $\\mathcal M$ and $f$ is an additive map from $\\mathcal M$ into its center vanishing on $[A,B]$ with $AB=0$; (2) $\\xi=0$, $L(I)\\in{\\mathcal Z}({\\mathcal M})$ and there exists an additive derivation $\\varphi$ such that $L(A)=\\varphi(A)+L(I)A$ for all $A$; (3) $\\xi=-1$, $L$ is a Jordan derivation; (4) $\\xi$ is rational and $\\xi\
T-Duality from super Lie n-algebra cocycles for super p-branes
Fiorenza, Domenico; Schreiber, Urs
2016-01-01
We compute the $L_\\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient $L_\\infty$-algebras are naturally related by an $L_\\infty$-isomorphism which we find to act on the super $p$-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. Moreover, we show that these $L_\\infty$-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". We find that the induced $L_\\infty$-extension is a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that this satisfies ...
su(2) Lie algebra approach for the Feynman propagator of the one-dimensional harmonic oscillator
Martínez, D.; Avendaño, C. G.
2014-04-01
We evaluate the Feynman propagator for the harmonic oscillator in one dimension. Considering the ladder operators for the Hamiltonian of this system, we construct a set of operators which satisfy the su(2) Lie algebra to obtain Mehler’s formula.
Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
Fu, Chih-Hao; Krasnov, Kirill
2017-01-01
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.
Lie algebraic analysis for the beam transport in the spherical electrostatic analyser
Institute of Scientific and Technical Information of China (English)
Lü Jian-Qin; Zhang Zhuo
2007-01-01
This paper uses the Lie algebraic method to analyse the charged particle trajectories in the spherical electrostatic analyser, and obtains the nonlinear solutions. The results show that the focusing abilities both in the x and y directions of the analyser are almost the same. Moreover, there exist dispersion effects in the x direction, and no dispersion effects in the y direction.
Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
Fu, Chih-Hao
2016-01-01
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbi...
A Lie-Algebra model for a noncommutative space time geometry
Doerfel, B D
2002-01-01
We propose a Lie-algebra model for noncommutative coordinate and momentum space . Based on a rigid commutation relation for the commutators of space time operators the model is quite constrained if one tries to keep Lorentz invariance as much as possible. We discuss the question of invariants esp. the definition of a mass.
Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method
Directory of Open Access Journals (Sweden)
Chein-Shan Liu
2013-01-01
Full Text Available We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.
Pairing Problem of Generators in Non-twisted Affine Lie Algebras
Institute of Scientific and Technical Information of China (English)
XU Hai-xia; LU Cai-hui
2001-01-01
In this paper, we discuss the pairing problem of generators in four affine Lie algebra. That is,for any given imaginary root vector x ∈ g (A), there exists y such that x and y generate a subalgebra containing g＇ (A).
Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability
Directory of Open Access Journals (Sweden)
Muhammad Ayub
2013-01-01
the case of k≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two kth-order (k≥3 ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, R. [Departamento de Geometria y Topologia, Facultat de CC. Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, E-28040 Madrid (Spain)]. E-mail: rutwig@mat.ucm.es
2007-03-12
Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint module. A criterion to decide whether a given deformation is invertible or not is given in dependence of the Poincare polynomial.
Solvable Lie algebras with an N-graded nilradical of maximal nilpotency degree and their invariants
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, R [I.M.I. and Dpto. Geometria y Topologia, Universidad Complutense de Madrid, Plaza de Ciencias, 3 E-28040 Madrid (Spain)], E-mail: rutwig@pdi.ucm.es
2010-04-09
The class of solvable Lie algebras with an N-graded nilradical of maximal nilpotency index is classified. It is shown that such solvable extensions are unique up to isomorphism. The generalized Casimir invariants for the N-graded nilradicals and their associated solvable extensions are computed by the method of moving frames.
Higher gauge theories from Lie n-algebras and off-shell covariantization
Energy Technology Data Exchange (ETDEWEB)
Carow-Watamura, Ursula; Heller, Marc Andre [Particle Theory and Cosmology Group, Department of Physics,Graduate School of Science, Tohoku University, Aoba-ku, Sendai 980-8578 (Japan); Ikeda, Noriaki [Department of Mathematical Sciences, Ritsumeikan University,Kusatsu, Shiga 525-8577 (Japan); Kaneko, Yukio; Watamura, Satoshi [Particle Theory and Cosmology Group, Department of Physics,Graduate School of Science, Tohoku University, Aoba-ku, Sendai 980-8578 (Japan)
2016-07-25
We analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we can obtain an off-shell covariantized higher gauge theory in 5 dimensions, which is similar to the one proposed in http://dx.doi.org/10.1007/JHEP09(2012)075.
Terzis, Petros A
2010-01-01
Lie group symmetry analysis for systems of coupled, nonlinear ordinary differential equations is performed in order to obtain the entire solution space to Einstein's field equations for vacuum Bianchi spacetime geometries. The symmetries used are the automorphisms of the Lie algebra of the corresponding three- dimensional isometry group acting on the hyper-surfaces of simultaneity for each Bianchi Type, as well as the scaling and the time reparametrization symmetry. The method is applied to Bianchi Types I; II; IV and V. The result is the acquisition, in each case, of the entire solution space of either Lorenzian of Euclidean signature. This includes all the known solutions for each Type and the general solution of Type IV (in terms of sixth Painlev\\'e transcendent PVI).
Maps Preserving Zero Lie Brackets on a Maximal Nilpotent Subalgebra of the Symplectic Algebra
Institute of Scientific and Technical Information of China (English)
Yan Xia ZHAO; Deng Yin WANG; Dong Fang JIA
2011-01-01
Let F be a field with char F ≠ 2,l a maximal nilpotent subalgebra of the symplectic algebra sp(2m,F).In this paper,we characterize linear maps of l which preserve zero Lie brackets in both directions.It is shown that for m ≥ 4,a map φ of l preserves zero Lie brackets in both directions if and only if φ = ψ cσToλαфdηf,where ψ c,σTo,λα,фd,ηf are the standard maps preserving zero Lie brackets in both directions.
Engel's Theorem of Jordan Lie Algebra and Its Applications%Jordan李代数的Engel定理及其应用
Institute of Scientific and Technical Information of China (English)
钱玲; 周佳; 陈良云
2012-01-01
证明了有限维Jordan李代数的Engel定理,并应用它得到了Jordan李代数的Cartan子代数的若干性质.%The authors prove Engel's theorem of Jordan Lie algebra and apply it to get some properties of Cartan subalgebras on Jordan Lie algebra.
Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups
Energy Technology Data Exchange (ETDEWEB)
Guedes, Carlos; Oriti, Daniele [Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam (Germany); Raasakka, Matti [Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam (Germany); LIPN, Institut Galilée, Université Paris-Nord, 99, av. Clement, 93430 Villetaneuse (France)
2013-08-15
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the conditions for its existence. In particular, the corresponding non-commutative star-product carried by this representation is obtained directly from the quantization map via deformation quantization. We then clarify under which conditions a unitary intertwiner between such algebra representation and the usual group representation can be constructed giving rise to the non-commutative plane waves and consequently, the non-commutative Fourier transform. The compact groups U(1) and SU(2) are considered for different choices of quantization maps, such as the symmetric and the Duflo map, and we exhibit the corresponding star-products, algebra representations, and non-commutative plane waves.
Astashkevich, Alexander
1999-01-01
This volume presents contributions by leading experts in the field. The articles are dedicated to D. B. Fuchs on the occasion of his 60th birthday. Contributors to the book were directly influenced by Professor Fuchs and include his students, friends, and professional colleagues. In addition to their research, they offer personal reminicences about Professor Fuchs, giving insight into the history of Russian mathematics. The main topics addressed in this unique work are infinite-dimensional Lie algebras with applications (vertex operator algebras, conformal field theory, quantum integrable syst
Gato-Rivera, Beatriz
2008-01-01
In 1998 the Adapted Ordering Method was developed for the study of the representation theory of the superconformal algebras in two dimensions. It allows: to determine the maximal dimension for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this talk I introduce the present version of the Adapted Ordering Method, published in J. Phys. A: Math. Theor. 41 (2008) 045201, which can be applied to general Lie algebras and superalgebras and their generalizations, provided they can be triangulated.
a Perspective on the Magic Square and the "special Unitary" Realization of Real Simple Lie Algebras
Santander, Mariano
2013-07-01
This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as "our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 𝕆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 𝕂 or tensor products 𝕂1 ⊗ 𝕂2, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ‧, ℍ‧, 𝕆‧ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 𝕂1, 𝕂2, which in all cases are obtained from ℝ, ℂ, ℂ‧, ℍ, ℍ‧, 𝕆, 𝕆‧ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.
Sasai, Yuya; Sasakura, Naoki
2009-12-01
We have investigated the unitarity of three dimensional noncommutative scalar field theory in Lie algebraic noncommutative spacetime [x̂i, x̂j] = 2iκɛijkx̂k, (i, j, k = 0, 1, 2). This noncommutative field theory possesses an SL(2, R)/Z2 group momentum space, which leads to a Hopf algebraic translational symmetry. We have checked the Cutkosky rule of the one-loop self-energy diagrams in the noncommutative φ3 theory when we include a braiding, which is necessary for the noncommutative field theory to possess the Hopf algebraic translational symmetry at quantum level. Then, we have found that the Cutkosky rule is satisfied if the mass of the scalar field is less than 1/√2κ , which however leads to be violations of the Cutkosky rule for smaller masses in more complicated diagrams.
Sasai, Yuya
2009-01-01
We investigate the unitarity of three dimensional noncommutative scalar field theory in the Lie algebraic noncommutative spacetime [x^i,x^j]=2i kappa epsilon^{ijk}x_k. This noncommutative field theory possesses a SL(2,R)/Z_2 group momentum space, which leads to a Hopf algebraic translational symmetry. We check the Cutkosky rule of the one-loop self-energy diagrams in the noncommutative phi^3 theory when we include a braiding, which is necessary for the noncommutative field theory to possess the Hopf algebraic translational symmetry at quantum level. Then, we find that the Cutkosky rule is satisfied if the mass is less than 1/(2^(1/2)kappa).
Classification of Hom-preLie Algebras in Dimension Two%二维Hom-preLie代数的分类
Institute of Scientific and Technical Information of China (English)
安慧辉; 康健; 王治淳
2014-01-01
In this paper, we mainly discuss the basic properties and the classification of Hom-Novikov algebras and Hom-preLie algebras in dimension two in the complex field. At first, we give the definition of the Hom-Novikov algebras, Hom-preLie algebras and some related defi-nitions. Then we discuss regular Hom-preLie algebras and give the necessary conditions for an Hom-preLie algebras to be pre-Lie type. We also give the direct sum of Hom-preLie alge-bras and get the necessary and sufficient condition for the existence of homomorphism between two Hom-preLie algebras. At last, with these definitions and their basic properties, we obtain the classification of the Hom-Novikov algebras and Hom-preLie algebras in two dimensions.%讨论了复数域上的二维Hom-Novikov 代数与Hom-preLie代数的基本性质以及分类。给出了Hom-Novikov 代数与Hom-preLie代数相关的一些基本定义和Hom-preLie是Pre-Lie型的必要条件；讨论Hom-preLie代数的直和，给出了两个Hom-preLie代数之间存在代数同态的充分必要条件。利用这些定义及其简单的性质，完成二维Hom-Novikov 代数与Hom-preLie代数的分类
Weighted Graph Theory Representation of Quantum Information Inspired by Lie Algebras
Belhaj, Abdelilah; Machkouri, Larbi; Sedra, Moulay Brahim; Ziti, Soumia
2016-01-01
Borrowing ideas from the relation between simply laced Lie algebras and Dynkin diagrams, a weighted graph theory representation of quantum information is addressed. In this way, the density matrix of a quantum state can be interpreted as a signless Laplacian matrix of an associated graph. Using similarities with root systems of simply laced Lie algebras, one-qubit theory is analyzed in some details and is found to be linked to a non-oriented weighted graph having two vertices. Moreover, this one-qubit theory is generalized to n-qubits. In this representation, quantum gates correspond to graph weight operations preserving the probability condition. A speculation from string theory, via D-brane quivers, is also given.
Zhu, Huangjun
2014-09-01
Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank 1. They are interesting originally because of their connection with rank-1 SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal informationally complete (IC) measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help us to understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.
A representation of Weyl-Heisenberg Lie algebra in the quaternionic setting
Muraleetharan, B.; Thirulogasanthar, K.; Sabadini, I.
2017-10-01
Using a left multiplication defined on a right quaternionic Hilbert space, linear self-adjoint momentum operators on a right quaternionic Hilbert space are defined in complete analogy with their complex counterpart. With the aid of the so-obtained position and momentum operators, we study the Heisenberg uncertainty principle on the whole set of quaternions and on a quaternionic slice, namely on a copy of the complex plane inside the quaternions. For the quaternionic harmonic oscillator, the uncertainty relation is shown to saturate on a neighborhood of the origin in the case we consider the whole set of quaternions, while it is saturated on the whole slice in the case we take the slice-wise approach. In analogy with the complex Weyl-Heisenberg Lie algebra, Lie algebraic structures are developed for the quaternionic case. Finally, we introduce a quaternionic displacement operator which is square integrable, irreducible and unitary, and we study its properties.
Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies
Energy Technology Data Exchange (ETDEWEB)
Bonora, L., E-mail: bonora@sissa.it [International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste (Italy); INFN, Sezione di Trieste (Italy); Bytsenko, A.A., E-mail: abyts@uel.br [Departamento de Fisica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina (Brazil)
2011-11-11
There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS{sub 3}, which also describe the three-dimensional Euclidean black holes, the pure N=1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2,Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.
PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
Guenther, Uwe
2010-01-01
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.
Kurnyavko, O. L.; Shirokov, I. V.
2016-07-01
We offer a method for constructing invariants of the coadjoint representation of Lie groups that reduces this problem to known problems of linear algebra. This method is based on passing to symplectic coordinates on the coadjoint representation orbits, which play the role of local coordinates on those orbits. The corresponding transition functions are their parametric equations. Eliminating the symplectic coordinates from the transition functions, we can obtain the complete set of invariants. The proposed method allows solving the problem of constructing invariants of the coadjoint representation for Lie groups with an arbitrary dimension and structure.
Type II chiral affine Lie algebras and string actions in doubled space
Hatsuda, Machiko; Siegel, Warren
2015-01-01
We present affine Lie algebras generated by the supercovariant derivatives and the supersymmetry generators for the left and right moving modes in the doubled space. Chirality is manifest in our doubled space as well as the T-duality symmetry. We present gauge invariant bosonic and superstring actions preserving the two-dimensional diffeomorphism invariance and the kappa-symmetry where doubled spacetime coordinates are chiral fields. The doubled space becomes the usual space by dimensional reduction constraints.
Non-linear Maps on Borel Subalgebras of Simple Lie Algebras Preserving Abelin Ideals
Institute of Scientific and Technical Information of China (English)
ZHAO Yan-xia; WANG Deng-yin
2012-01-01
Let g be a complex simple Lie algebra of rank l,b the standard Borel subalgebra.An invertible map on b is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension.In this article,by using some results of Chevalley groups,the theory of root systems and root space decomposition,the author gives an explicit description on such maps of b.
Pricing multi-asset financial derivatives with time-dependent parameters—Lie algebraic approach
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C. F. Lo
2002-01-01
Full Text Available We present a Lie algebraic technique for the valuation of multi-asset financial derivatives with time-dependent parameters. Exploiting the dynamical symmetry of the pricing partial differential equations of the financial derivatives, the new method enables us to derive analytical closed-form pricing formulae very straightforwardly. We believe that this new approach will provide an efficient and easy-to-use method for the valuation of financial derivatives.
Institute of Scientific and Technical Information of China (English)
ZHANG Zhuo; L(U) Jian-Qin
2008-01-01
In this paper, the nonlinear transport of intense bunched beams in electrostatic quadrupoles is analyzed using the Lie algebraic method, and the results are briefly presented of the linear matrix approximation and the second order correction of particle trajectory in the state space. Beam having K-V distribution and Gaussian distribution approximation are respectively considered. A brief discussion is also given of the total effects of the quadrupole and the space charge forces on the evolution of the beam envelope.
The Wheeler-DeWitt Equation in Filćhenkov Model: The Lie Algebraic Approach
Panahi, H.; Zarrinkamar, S.; Baradaran, M.
2016-11-01
The Wheeler-DeWitt equation in Filćhenkov model with terms related to strings, dust, relativistic matter, bosons and fermions, and ultra stiff matter is solved in a quasi-exact analytical manner via the Lie algebraic approach. In the calculations, using the representation theory of sl(2), the general (N+1)-dimensional matrix equation is constructed whose determinant yields the solutions of the problem.
Indecomposable representations of the Lie algebra of derivations for d-torus
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let DerA be the Lie algebra of derivations of the d-torus A = C[t1± 1, . . . , td±1]. By applying Shen-Larsson’s functors we get a class of indecomposable DerA-modules from finite-dimensional indecomposable gld-modules. We also give a complete description of the submodules of these indecomposable DerA-modules. Our results generalize those obtained by Rao.
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Nader Ali Makboul Hassan
2014-01-01
Full Text Available This paper is an attempt to stress the usefulness of the multi-variable special functions. In this paper, we derive certain generating relations involving 2-indices 5-variables 5-parameters Tricomi functions (2I5V5PTF by using a Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.
McRae, Robert
2016-08-01
For a finite-dimensional simple Lie algebra {{g}}, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra {{widehat{{g}}}} at a fixed level {ℓin{N}} with a certain tensor category of finite-dimensional {{g}}-modules. More precisely, the category of level ℓ standard {{widehat{{g}}}}-modules is the module category for the simple vertex operator algebra {L_{widehat{{g}}}(ℓ, 0)}, and as is well known, this category is equivalent as an abelian category to {{D}({g},ℓ)}, the category of finite-dimensional modules for the Zhu's algebra {A{(L_{widehat{{g}}}(ℓ, 0))}}, which is a quotient of {U({g})}. Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in {{D}({g},ℓ)} induced from the associativity isomorphisms constructed by Huang and Lepowsky in {{L_{widehat{{g}}}(ℓ, 0) - {mod}}}. This construction shows that {{D}({g},ℓ)} is closely related to the Drinfeld category of {U({g})}[[h
Institute of Scientific and Technical Information of China (English)
MENG; Qingtian
2001-01-01
［1］Iachello, F, Levine, R. D., Algebraic approach to molecular rotation-vibration spectra, I. Diatomic molecules, J, Chem.Phys.. 1982, 77: 3046.［2］Iachello. F.. Oss, S., Overtone frequencies and intensities of bent XY2 molecules in the vibron model, J. Mol. Spectrosc.,1990,142: 85.［3］Van Roosmalen, O. S., Iachello, F., Levine, R. D. et al., Algebraic approach to molecular rotation-vibration spectra, II. Triatomic molecules, J. Chem. Phys., 1983, 79: 2515.［4］Iachello, F., Levine, R. D., Algebraic approach to molecular rotation-vibration spectra, Int. J. Quantum Chem., 1983, 23:1679.［5］Cooper, I. L., Levine, R. D., Computed overtone spectra of linear triatomic molecules by dynamical symmetry, J. Mol. Spectrosc., 1991, 148: 391.［6］Iachello. F., Manini. N., Oss, S., Quasi-linear four-atomic molecules in the vibron model, J. Mol. Spectrosc., 1992, 156:190.［7］Wiesenfeld, L.. The vibron model for methane: stretch-bend interactions, J. Mol. Spectrosc., 1997, 184: 277.［8］Zheng, Y.. Ding, S., Vibrational spectra of HCN and OCS from second-order expansion of the U1(4) U2(4) algebra,Phys. Lett. A. 1999. 256: 197.［9］Zheng, Y.. Ding. S., Algebraic method for determining the potential energy surface for nonlinear triatomic molecules, Chem. Phys., 1999, 247: 225.［10］Zheng, Y.. Ding, S.. Algebraic description of stretching and bending vibrational spectra of H2O and H2S, J. Mol. Spectrosc.,2000. 201: 109.［11］Meng. Q., Zheng, Y., Ding, S., Lie algebraic approach to Fermi resonance levels of CS2 and CO2, Int. J. Quantum Chem.,2001, 81: 154.［12］Ding, S., Zheng, Y., Lie algebraic approach to potential energy surface for symmetric triatomic molecules, J. Chem. Phys.,1999. 111: 4466.［13］Zheng. Y., Ding, S., Algebraic approach to the potential energy surface for the electronic ground state of ozone, Chem.Phys.. 2000. 255: 217.［14］Zheng. Y., Ding, S., Theoretical study of nonlinear triatomic molecular potential energy surfaces: Lie
Engel’ s theorem for generalized H-Lie algebras%广义H-李代数的Engel定理
Institute of Scientific and Technical Information of China (English)
郭双建; 董丽红
2014-01-01
It is studied that the representation of the Lie algebras in the Yetter-Drinfel’ d category HH YD( i.e.generalized H-Lie algebras) .The Engel’ s theorem for generalized H-Lie algebras is proved:Let L be a generalized H-Lie algebra, if every cyclic Yetter-Drinfel’ d submodule of L is ad-nilpotent, then L is nilpotent.%研究了Yetter-Drinfel’ d范畴HH YD中李代数（即广义H-李代数）的表示，证明了广义H-李代数的Engel定理：设L是一个广义H-李代数，如果L的每一个循环Yetter-Drinfel’ d模都是ad-幂零的，那么L是幂零的。
Das, A; Das, A; Wotzasek, C
1995-01-01
We study a supersymmetric 2-dimensional harmonic oscillator which carries a representation of the general graded Lie algebra GL(2\\vert1) formulate it on the superspace, and discuss its physical spectrum.
Maximal Abelian subalgebras of pseudoeuclidean real Lie algebras and their application in physics
Thomova, Zora
1998-12-01
We construct the conjugacy classes of maximal abelian subalgebras (MASAs) of the real pseudoeuclidean Lie algebras e(p, q) under the conjugation by the corresponding pseudoeuclidean Lie groups E(p, q). The algebra e( p, q) is a semi-direct sum of the pseudoorthogonal algebra o(p, q) and the abelian ideal of translations T(p + q). We use this particular structure to construct first the splitting MASAs, which are themselves direct sums of subalgebras of o(p, q) and T(p + q). Splitting MASAs give rise to the nonsplitting MASAs of e(p, q). The results for q = 0, 1 and 2 are entirely explicit. MASAs of e(p, 0) and e( p, 1) are used to construct conformally nonequivalent coordinate systems in which the wave equation and Hamilton-Jacobi equations allow the separation of variables. As an application of subgroup classification we perform symmetry reduction for two nonlinear partial differential equations. The method of symmetry reduction is used to obtain analytical solutions of the Landau-Lifshitz and a nonlinear diffusion equations. The symmetry group is found for both equations and all two-dimensional subgroups are classified. These are used to reduce both equations to ordinary differential equations, which are solved in terms of elliptic functions.
Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
Energy Technology Data Exchange (ETDEWEB)
Sakai, Kazumitsu, E-mail: sakai@gokutan.c.u-tokyo.ac.jp [Institute of Physics, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902 (Japan)
2013-02-11
We provide multiple Schramm-Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE traces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. As a concrete example, we apply the formula to the su{sup -hat} (2){sub k}-WZW model. Utilizing the fusion rules in the model, we conjecture that there exists a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Furthermore, solving the Knizhnik-Zamolodchikov equation, we exactly compute the probabilities of occurrence for certain configurations (i.e. crossing probabilities) of traces for the triple SLE.
Lie Ideals in AF C*-Algebras%AF C*-代数中的Lie理想
Institute of Scientific and Technical Information of China (English)
纪培胜; 王琳
2005-01-01
本文描述了AF C*-代数中闭Lie理想,证明了如果AF C*-代数A中的线性流形L是A的闭Lie理想,则存在A的闭结合理想I和A的典型masa D中的闭子代数研使得([A, I])(∪)L(∪)I+EI,并且A中每一个这种形式的闭子空间都是A的闭Lie理想.%We study Lie ideals in unital AF C*-algebras. It is shown that if a linear manifold L in an AF C*-algebra A is a closed Lie ideal in A, then there exists a closed associative ideal I and a closed subalgebra EI of the canonical masa D of A such that ([A, I])(∪)L(∪)I+EI,andthat every closed subspace in this form is a closed Lie ideal in A.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The Hamiltonian describing rotational spectra of linear triatomic molecules has been derived by using the dynamical Lie algebra of symmetry group U1(4)U,(4). After rovibrational interactions being considered, the eigenvalue expression of the Hamiltonian has the form of term value equation commonly used in spectrum analysis. The molecular rotational constants can be obtained by using the expression and fitting it to the observed lines. As an example, the rotational levels of v2 band for transition (0200-0110) of molecules N2O and HCN have been fitted and the fitting root-mean-square errors (RMS) are 0.00001 and 0.0014 cm-1, respectively.
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Mohammad Shahzad
2016-05-01
Full Text Available This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM by State Space Exact Linearization (SSEL technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustness of the technique implemented on the chosen chaotic system.
Lie algebraic analysis for the nonlinear transport of intense pulsed beams in electrostatics lenses
Institute of Scientific and Technical Information of China (English)
Lu Jian-Qin; Li Jin-Hai
2004-01-01
The Lie algebraic method is applied to the analysis of the nonlinear transport of an intense pulsed beam in cylindrically symmetrical electrostatic lenses, and particle orbits in a six-dimensional phase space (x, px, y, py, τ, pτ)are obtained in the second order approximation. They can also be acquired in the third or higher order approximation if needed. In the analysis, we divide the electrostatic lenses into several segments. Each segment is considered as a uniform accelerating field, and each dividing point is treated as a thin lens. The particle distribution in a three-dimensional ellipsoid is of Gaussian type.
Kegel theorem for generalized Lie algebras%广义Lie代数的Kegel定理
Institute of Scientific and Technical Information of China (English)
陈华喜; 张崔斌; 董丽红
2014-01-01
设π是一个群，（H，σ）是一个余三角Hopfπ-余代数，在π-H-余模范畴中构造了一类广义Lie代数，并且得到了经典的Kegel定理。%Letπbe a group and (H,σ)a cotriangular Hopfπ-coalgebra.The difinitions of a class of generalized Lie algebras in the category of rightπ-comodules over H are introduced,and an analogue of the classical Kegels theorem is obtained.
Bezrukavnikov, Roman
2010-01-01
We prove most of Lusztig's conjectures from the paper "Bases in equivariant K-theory II", including the existence of a canonical basis in the Grothendieck group of a Springer fiber. The conjectures also predict that this basis controls numerics of representations of the Lie algebra of a semi-simple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a non-commutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is shown to be compatible with the positive characteristic of the Beilinson-Bernstein localization equivalences. On the other hand, it is compatible with the t-structure arising from the equivalence of Arkhipov-Bezrukavnikov with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group which was inspired by local geometric Langlands duality. This allows one to apply Frobenius p...
Fermionic realisations of simple Lie algebras and their invariant fermionic operators
Azcarraga, J A D
2000-01-01
We study the representation D of a simple compact Lie algebra g of rank l constructed with the aid of the hermitian Dirac matrices of a ( dim g )-dimensional euclidean space. The irreducible representations of g contained in D are found by providing a general construction on suitable fermionic Fock spaces. We give full details not only for the simplest odd and even cases, namely su(2) and su(3) , but also for the next ( dim g )-even case of su(5) . Our results are far reaching: they apply to any g -invariant quantum mechanical system containing dim g fermions. Another reason for undertaking this study is to examine the role of the g -invariant fermionic operators that naturally arise. These are given in terms of products of an odd number of gamma matrices, and include, besides a cubic operator, l-1 fermionic scalars of higher order. The latter are constructed from the Lie algebra cohomology cocycles, and must be considered to be of theoretical significance similar to the cubic operator. In the ( dim g )-even ...
Langlois, Michel
2014-01-01
In order to generalize the relativistic notion of boost to the case of non inertial particles and to general relativity, we come back to the definition of Lie group of Lorentz matrices and its Lie algebra and we study how this group acts on the Minskowski space. We thus define the notion of tangent boost along a worldline. This notion very general notion gives a useful tool both in special relativity (for non inertial particles or/and for non rectilinear coordinates) and in general relativity. We also introduce a matrix of the Lie algebra which, together with the tangent boost, gives the whole dynamical description of the considered system (acceleration and Thomas rotation). After studying the properties of Lie algebra matrices and of their reduced forms, we show that the Lie group of special Lorentz matrices has four one-parameter subgroups. These tools lead us to introduce the Thomas rotation in a quite general way. At the end of the paper, we present some examples using these tools and we consider the case...
Spectral theory and nonlinear problems Théorie spectrale et problèmes non-linéaires
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Ahmed Lesfari
2010-06-01
Full Text Available We present a Lie algebra theoretical schema leading to integrable systems, based on the Kostant-Kirillov coadjoint action. Many problems on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras yield large classes of extended Lax pairs. A general statement leading to such situations is given by the Adler-Kostant-Symes theorem and the van Moerbeke-Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety (or some subabelian variety of it of the spectral curve. The complex flows generated by the constants of the motion are straight line motions on these varieties. We study the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view and their relation with the Kac-Moody extension of some algebras. We will present in detail the Griffith's aproach and his cohomological interpretation of linearization test for solving integrable systems without reference to Kac-Moody algebras. We will discuss several examples of integrable systems of relevance in mathematical physics.
L1-determined ideals in group algebras of exponential Lie groups
Ungermann, Oliver
2012-01-01
A locally compact group $G$ is said to be $\\ast$-regular if the natural map $\\Psi:\\Prim C^\\ast(G)\\to\\Prim_{\\ast} L^1(G)$ is a homeomorphism with respect to the Jacobson topologies on the primitive ideal spaces $\\Prim C^\\ast(G)$ and $\\Prim_{\\ast} L^1(G)$. In 1980 J. Boidol characterized the $\\ast$-regular ones among all exponential Lie groups by a purely algebraic condition. In this article we introduce the notion of $L^1$-determined ideals in order to discuss the weaker property of primitive $\\ast$-regularity. We give two sufficient criteria for closed ideals $I$ of $C^\\ast(G)$ to be $L^1$-determined. Herefrom we deduce a strategy to prove that a given exponential Lie group is primitive $\\ast$-regular. The author proved in his thesis that all exponential Lie groups of dimension $\\le 7$ have this property. So far no counter-example is known. Here we discuss the example $G=B_5$, the only critical one in dimension $\\le 5$.
Frattini Subalgebras and Nonimbedding Theorem of n-Lie Algebras%n-Lie代数的Frattini子代数及非嵌入定理
Institute of Scientific and Technical Information of China (English)
白瑞蒲; 周和月; 刘学文
2006-01-01
In this paper, we prove the nonimbedding theorem in nilpotent n-Lie algebras which is an analogue to the nonimbedding theorem of Burnsids in groups of prime power order. We also study the properties of Frattini suialgebras of n-Lie algebras over the field with characteristic zero, and prove that the Frattini subalgebra of any k-solvable (k≥2) n-Lie algebra is zero.
Akbarzadeh, Rasoul
2016-01-01
In 2001, A. V. Borisov, I. S. Mamaev, and V. V. Sokolov discovered a new integrable case on the Lie algebra so(4). This is a Hamiltonian system with two degrees of freedom, where both the Hamiltonian and the additional integral are homogenous polynomials of degrees 2 and 4, respectively. In this paper, the topology of isoenergy surfaces for the integrable case under consideration on the Lie algebra so(4) and the critical points of the Hamiltonian under consideration for different values of parameters are described and the bifurcation values of the Hamiltonian are constructed. Also, a description of bifurcation complexes and typical forms of the bifurcation diagram of the system are presented.
Analytical Lie-algebraic solution of a 3D sound propagation problem in the ocean
Energy Technology Data Exchange (ETDEWEB)
Petrov, P.S., E-mail: petrov@poi.dvo.ru [Il' ichev Pacific Oceanological Institute, 43 Baltiyskaya str., Vladivostok, 690041 (Russian Federation); Prants, S.V., E-mail: prants@poi.dvo.ru [Il' ichev Pacific Oceanological Institute, 43 Baltiyskaya str., Vladivostok, 690041 (Russian Federation); Petrova, T.N., E-mail: petrova.tn@dvfu.ru [Far Eastern Federal University, 8 Sukhanova str., 690950, Vladivostok (Russian Federation)
2017-06-21
The problem of sound propagation in a shallow sea with variable bottom slope is considered. The sound pressure field produced by a time-harmonic point source in such inhomogeneous 3D waveguide is expressed in the form of a modal expansion. The expansion coefficients are computed using the adiabatic mode parabolic equation theory. The mode parabolic equations are solved explicitly, and the analytical expressions for the modal coefficients are obtained using a Lie-algebraic technique. - Highlights: • A group-theoretical approach is applied to a problem of sound propagation in a shallow sea with variable bottom slope. • An analytical solution of this problem is obtained in the form of modal expansion with analytical expressions of the coefficients. • Our result is the only analytical solution of the 3D sound propagation problem with no translational invariance. • This solution can be used for the validation of the numerical propagation models.
Studies of Rigid Rotor-Rigid Surface Scattering in Dynamical Lie Algebraic Method
Institute of Scientific and Technical Information of China (English)
WANG Xiao-Yan; DING Shi-Liang
2004-01-01
The dynamical Lie algebraic method is used for the description of statistical mechanics of rotationally inelastic molecule-surface scattering. It can give the time-evolution operators about the low power of a+ and a by solving a set of coupled nonlinear differential equations. For considering the contribution of the high power of a+ and a, we use the Magnus formula. Thus, with the time-evolution operators we can get the statistical average values of the measurable quantities in terms of the density operator formalism in statistical mechanics. The method is applied to the scattering of N2 (rigid rotor) by a flat, rigid surface to illustrate its general procedure. The results demonstrate that the method is useful for describing the statistical dynamics of gas-surface scattering.
Non-Linear Integral Equations for complex Affine Toda associated to simply laced Lie algebras
Zinn-Justin, P
1998-01-01
A set of coupled non-linear integral equations is derived for a class of models connected with the quantum group $U_q(\\hat g)$ ($q=e^{i\\gamma}$ and $g$ simply laced Lie algebra), which are solvable using the Bethe Ansatz; these equations describe arbitrary excited states of a system with finite spatial length $L$. They generalize the Destri-De Vega equation for the Sine-Gordon/massive Thirring model to affine Toda field theory with imaginary coupling constant. As an application, the central charge and all the conformal weights of the UV conformal field theory are extracted in a straightforward manner. The quantum group truncation for rational values of $\\gamma/\\pi$ is discussed in detail; in the UV limit we recover through this procedure the RCFTs with extended $W(g)$ conformal symmetry.
Automorphism Group of Heisenberg Jordan-Lie Algebra%Heisenberg Jordan-Lie代数的自同构群
Institute of Scientific and Technical Information of China (English)
周佳
2014-01-01
We introduced the notion of Heisenberg Jordan-Lie algebra so as to investigate some subgroups of the automorphism group Aut(H)of Heisenberg Jordan-Lie algebra H.Moreover,we discussed some basic structure of the automorphism group Aut (H ) in the case of H being low-dimensional.%通过给出 Heisenberg Jordan-Lie 代数的定义，得到 Heisenberg Jordan-Lie 代数H 的自同构群Aut(H )的一些子群，并在 H 为低维的情形下，讨论了自同构群 Aut (H )的基本结构。
BKM Lie superalgebras from counting twisted CHL dyons
Govindarajan, Suresh
2011-05-01
Following Sen, we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with mathcal{N} = 4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M 24, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to a periodic table of BKM Lie superalgebras with properties that are consistent with physical expectations.
Graphical Tensor Product Reduction Scheme for the Lie Algebras so(5) = sp(2), su(3), and g(2)
Vlasii, N D; Wiese, U -J
2015-01-01
We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5) = sp(2), su(3), and g(2). This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a "landscape" of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic "girdle" method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.
Graphical tensor product reduction scheme for the Lie algebras so(5) = sp(2) , su(3) , and g(2)
Vlasii, N. D.; von Rütte, F.; Wiese, U.-J.
2016-08-01
We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5) = sp(2) , su(3) , and g(2) . This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a "landscape" of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic "girdle" method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.
Institute of Scientific and Technical Information of China (English)
白瑞蒲; 程宇; 李佳倩; 孟伟
2014-01-01
3-Lie algebras have close relationships with many important fields in mathemat-ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char-acteristic zero are provided.
Emergence of the world with Lie-N-algebra and M-dimensions from nothing
Sepehri, Alireza
2016-01-01
In this paper, we propose a new model in Lie-N-algebra that removes big bang singularity and produces the world with all it's objects and dimensions from nothing. We name this theory as G(God)-theory. In this model, first, two types of energies with opposite signs are produced from nothing such as the sum over them be zero. They create two types of branes with opposite quantum numbers which interact with each other by exchanging bosonic tensor fields like graviton and compact. By compacting branes, fermionic tensor fields are emerged which some of them play the role of the gravitinos. Also, some dimensions take extra (i) factors, their properties become different and they behave like time dimensions. Gravitons and gravitinos create two types of wormholes which lead to the oscillation of branes between expansion and contracting branches. These wormholes produce a repulsive gravity in compacted branes and cause that their particles get away from each other and expansion branch begin. Also, they create an attrac...
Generalized Toda Mechanics Associated with Classical Lie Algebras and Their Reductions
Institute of Scientific and Technical Information of China (English)
ZHAO Liu; LIU Wang-Yun; YANG Zhan-Ying
2004-01-01
For any classical Lie algebra g, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers (m, n). The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets areprovided, and explicit examples for g = Br, Cr,Dr with m, n ≤ 3 are also given. For all m, n, it is shown that the dynamics of the (m, n - 1)- and the (m - 1, n)-Toda chains are natural reductions of that of the (m, n)-chain,and for m = n, there is also a family of symmetrically reduced Toda systems, the (m, m)sym-Toda systems, which are also integrable. In the quantum case, all (m,n)-Toda systems with m ＞ 1 or n ＞ 1 describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all (m, n)-Toda systems survive after quantization.
Coproduct and star product in field theories on Lie-algebra noncommutative space-times
Amelino-Camelia, Giovanni; Arzano, Michele
2002-04-01
We propose a new approach to field theory on κ-Minkowski noncommutative space-time, a popular example of Lie-algebra space-time. Our proposal is essentially based on the introduction of a star product, a technique which is proving to be very fruitful in analogous studies of canonical noncommutative space-times, such as the ones recently found to play a role in the description of certain string-theory backgrounds. We find to be incorrect the expectation, previously reported in the literature, that the lack of symmetry of the κ-Poincaré coproduct should lead to interaction vertices that are not symmetric under exchanges of the momenta of identical particles entering the relevant processes. We show that in κ-Minkowski the coproduct and the star product must indeed treat momenta in a nonsymmetric way, but the overall structure of interaction vertices is symmetric under exchange of identical particles. We also show that in κ-Minkowski field theories it is convenient to introduce the concepts of ``planar'' and ``nonplanar'' Feynman loop diagrams, again in close analogy with the corresponding concepts previously introduced in the study of field theories in canonical noncommutative space-times.
Tang, Wanjie; Li, Bin; Huang, Xiaoqi; Jiang, Xiaoyu; Li, Fei; Wang, Lijuan; Chen, Taolin; Wang, Jinhui; Gong, Qiyong; Yang, Yanchun
2013-10-01
Few studies have used neuroimaging to characterize treatment-refractory obsessive-compulsive disorder (OCD). This study sought to explore gray matter structure in patients with treatment-refractory OCD and compare it with that of healthy controls. A total of 18 subjects with treatment-refractory OCD and 26 healthy volunteers were analyzed by MRI using a 3.0-T scanner and voxel-based morphometry (VBM). Diffeomorphic anatomical registration using exponentiated Lie algebra (DARTEL) was used to identify structural changes in gray matter associated with treatment-refractory OCD. A partial correlation model was used to analyze whether morphometric changes were associated with Yale-Brown Obsessive-Compulsive Scale scores and illness duration. Gray matter volume did not differ significantly between the two groups. Treatment-refractory OCD patients showed significantly lower gray matter density than healthy subjects in the left posterior cingulate cortex (PCC) and mediodorsal thalamus (MD) and significantly higher gray matter density in the left dorsal striatum (putamen). These changes did not correlate with symptom severity or illness duration. Our findings provide new evidence of deficits in gray matter density in treatment-refractory OCD patients. These patients may show characteristic density abnormalities in the left PCC, MD and dorsal striatum (putamen), which should be verified in longitudinal studies. © 2013. Published by Elsevier Inc. All rights reserved.
New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.
Institute of Scientific and Technical Information of China (English)
薛胜利
2013-01-01
由于域上的典型线性李代数都是单李代数，而交换幺环上的典型线性李代数未必仍为单李代数，而单李代数的结构分类对研究半单纯李代数的结构分类，以及可解、幂零李代数的研究至关重要，在这里我们得出了交换幺环上典型线性李代数为R-单李代数的充要条件。%Because the classical linear Lie algebra over a field is a simple Lie algebra, but the classical linear Lie algebra over a ring with identity may not still be a simple Lie algebra, and the classification and structure of simple Lie algebras is critical to the study of the the structure and classification of the semi-simple Lie algebra, even solvable, nilpotent Lie algebra, here we give the necessary and sufficient condition on which the R-classical linear Lie algebra is R-simple Lie algebra.
BKM Lie superalgebras from counting twisted CHL dyons
Govindarajan, Suresh
2010-01-01
Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko[arXiv:0812.3962]. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This lead...
Institute of Scientific and Technical Information of China (English)
夏铁成; 李季
2008-01-01
Based on the generalization of Lie algebra An-1,two types of new Lie algebras were worked out and the integrability of the related hierarchies of evolution equations were proved in the sense of Liouville.
Institute of Scientific and Technical Information of China (English)
ZHAO xiao-Song; L(U) Jian-Qin
2009-01-01
Both the PIC(Particle-In-Cell) model and the Lie algebraic method can be used to simulate the transport of intense continuous beams.The PIC model is to calculate the space charge field,which is blended into the external field,and then simulate the trajectories of particles in the total field;the Lie algebraic method is to simulate the intense continuous beam transport with transport matrixes.Two simulation codes based on the two methods are developed respectively,and the simulated results of transport in a set of electrostatic lenses are compared.It is found that the results from the two codes are in agreement with each other.and both approaches have their own merits.
Energy Technology Data Exchange (ETDEWEB)
Kozlov, I K [M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow (Russian Federation)
2014-04-30
In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra so(4), which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank 0 are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra so(4) by taking a natural limit. Bibliography: 21 titles.
Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models
Directory of Open Access Journals (Sweden)
C. F. Lo
2013-01-01
Full Text Available The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely, the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.
Akbarzadeh, Rasoul; Haghighatdoost, Ghorbanali
2015-05-01
In 2001, A.V. Borisov, I. S.Mamaev, and V.V. Sokolov discovered a new integrable case on the Lie algebra so(4). This system coincides with the Poincaré equations on the Lie algebra so(4), which describe the motion of a body with cavities filled with an incompressible vortex fluid. Moreover, the Poincaré equations describe the motion of a four-dimensional gyroscope. In this paper topological properties of this system are studied. In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fomenko invariants are calculated. Thereby, a classification of isoenergy surfaces for this system up to the rough Liouville equivalence is obtained.
Some properties of the intersections of maximal subalgebras in Lie color algebras%李Color代数极大子代数的基本性质
Institute of Scientific and Technical Information of China (English)
宋华; 王晨迪
2012-01-01
In this paper,we develop initially the theory on the intersections of maximal subalgebras for Lie color algebras,obtain their some properties and give some necessary and sufficient conditions for solvable Lie color algebras and nilpotent Lie color algebras,respectively.%主要把Frattini子代数的性质推广到李Color代数,得到了它们的若干性质,并利用其性质分别给出可解和幂零李Color代数的几个充分必要条件.
Automorphism Group of a Class of Heisenberg n-Lie Algebras%一类Heisenberg n-李代数的自同构群
Institute of Scientific and Technical Information of China (English)
白瑞蒲; 刘丽丽
2011-01-01
本文主要研究Heisenberg n-李代数的结构.给出了一类(3m+1)-维Heisenberg 3-李代数及(nm+1)-维Heisenberg n-李代数的自同构群.且给出了自同构的具体表达式.%This paper mainly concerns Heisenberg n-Lie algebras. The structure of automorphism groups of (3m+1)-dimensional Heisenberg 3-Lie algebras is determined. The automorphism groups of (mn+1)-dimensional Heisenberg n-Lie algebras are studied; the concrete expression of every automorphism is given.
Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras
Park, Jae-Suk
2015-01-01
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in t...
A new matrix method for the Casimir operators of the Lie algebras wsp(N,R) and Isp(2N,R)
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, Rutwig [Dpto. Geometria y Topologia, Fac. CC. Matematicas, Universidad Complutense de Madrid, Ciudad Universitaria s/n, E-28040 Madrid (Spain)
2005-05-13
A method is given to determine the Casimir operators of the perfect Lie algebras wsp(N,R) = sp(2N,R) +-vector {sub {gamma}}{sub {omega}{sub 1}}{sub +{gamma}{sub 0}} h{sub N} and the inhomogeneous Lie algebras Isp(2N,R) in terms of polynomials associated with a parametrized (2N + 1) x (2N + 1)-matrix. For the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation. We further analyse the invariants for the extended Schroedinger algebra S-circumflex(N) in (N + 1) dimensions, which arises naturally as a subalgebra of wsp(N,R). The method is extended to other classes of Lie algebras, and some applications to the missing label problem are given.
Affine Mirkovi\\'c-Vilonen polytopes
Baumann, Pierre; Tingley, Peter
2011-01-01
Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by B(-\\infty), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirkovi\\'c-Vilonen polytope, can be associated to each element in B(-\\infty). This polytope sits in the dual space of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In this paper, we generalize this construction to the case where g is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root \\delta. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool...
Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.
2016-08-01
We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.
de Azcárraga, J A; Picon, M; Varela, O; Azcarraga, Jose A. de; Izquierdo, Jose M.; Picon, Moises; Varela, Oscar
2003-01-01
We study how to generate new Lie algebras $\\mathcal{G}(N_0,..., N_p,...,N_n)$ from a given one $\\mathcal{G}$. The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter $\\lambda$ which rescales the coordinates of the Lie (super)group $G$, $g^{i_p} \\to \\lambda^p g^{i_p}$, in a way subordinated to the splitting of $\\mathcal{G}$ as a sum $V_0 \\oplus ... \\oplus V_p \\oplus ... \\oplus V_n$ of vector subspaces. We also show that, under certain conditions, one of the obtained algebras may correspond to a generalized \\.In\\"on\\"u-Wigner contraction in the sense of Weimar-Woods, but not in general. The method is used to derive the M-theory superalgebra, including its Lorentz part, from $osp(1|32)$. It is also extended to include gauge free differential (super)algebras and Chern-Simons theories, and then applied to D=3 CS supergravity.
The Hom-structures on Filiform Lie algebras Qn%Filiform李代数Qn的Hom-结构
Institute of Scientific and Technical Information of China (English)
于欢欢; 刘文德
2015-01-01
In this paper, we prove that a linear operator on a finite-dimensional Z-graded Lie algebra is a Hom-structure if and only if its homogeneous components are Hom-structures. We also compute homogeneous Hom-structures on a finite dimensional Z-graded Filiform Lie algebra Qn over an algebraically closed field of characteristic zero. As a consequence, we determine all the Hom-structures on Qn.%首先证明了有限维Z-阶化李代数上的一个线性算子是Hom-结构的充分必要条件，即它的每个齐次分支也是Hom-结构。然后计算了特征零代数闭域上一类有限维Z-阶化Filiform李代数Qn 的齐次Hom-结构，从而决定了Qn 的所有Hom-结构。
Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields
Fiorenza, Domenico; Schreiber, Urs
2013-01-01
We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type sigma-model branes (open brane ending on background brane) are encoded precisely in (super-) L-infinity-extension theory and how the resulting "extended (super-)spacetimes" formalize spacetimes containing sigma model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super p-brane spectrum of superstring/M-theory is realized this way, including the pure sigma-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional spacetime with an M2-brane condensate turns out to be the ...
BKM Lie superalgebras from dyon spectra in Z_N CHL orbifolds for composite N
Govindarajan, Suresh
2009-01-01
We show that the generating function of electrically charged 1/2-BPS states in N=4 supersymmetric CHL Z_N orbifolds of the heterotic string on T^6 are given by multiplicative eta-products. The eta-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/0609109] for Z_N orbifolds when N is non-prime. We study the Z_4 CHL orbifold in detail and show that the associated Siegel modular forms, \\Phi_3(Z) and \\widetilde{\\Phi}_3(Z), are given by the square of the product of three even genus-two theta constants. Extending work by us[arXiv:0807.4451] as well as Cheng and Dabholkar[arXiv:0809.4258], we show that their `square roots' appear as the denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie superalgebras. The BKM Lie superalgebra associated with the generating function of 1/4-BPS states, i.e., \\widetilde{\\Phi}_3(Z) h...
Institute of Scientific and Technical Information of China (English)
关大任; 易希璋; 丁世良; 郑雨军; 刘建勇
1999-01-01
A dynamical Lie algebraic method has been applied to treating the quantum dynamics of dissociative adsorption of H2 on a static flat metal surface. An LEPS potential energy surface has been used to describe the interaction of H2 with Ni（100） surface. The dependence of the initial state-selected dissociation probability was obtained analytically on the initial kinetic energy and time. A comparison with other theoretical calculations and experiments is made. The results show that the method can be effectively used to describe the dynamics of reactive gas-surface scattering.
Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1) Dimensions
Institute of Scientific and Technical Information of China (English)
ZHAO Xue-Qing; L(U)Jing-Fa
2004-01-01
The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.
Matone, Marco
2016-11-01
Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp (X) exp (Y)=exp (W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp (X) exp (Y) exp (Z)=exp (W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.
Energy Technology Data Exchange (ETDEWEB)
Matone, Marco [Universita di Padova, Dipartimento di Fisica e Astronomia ' ' G. Galilei' ' , Padua (Italy); Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padua (Italy)
2016-11-15
Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp(X) exp(Y) = exp(W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp(X) exp(Y) exp(Z) = exp(W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper. (orig.)
Dp-branes, NS5-branes and U-duality from nonabelian (2,0) theory with Lie 3-algebra
Honma, Yoshinori; Shiba, Shotaro
2011-01-01
We derive the super Yang-Mills action of Dp-branes on a torus T^{p-4} from the nonabelian (2,0) theory with Lie 3-algebra. Our realization is based on Lie 3-algebra with pairs of Lorentzian metric generators. The resultant theory then has negative norm modes, but it results in a unitary theory by setting VEV's of these modes. This procedure corresponds to the torus compactification, therefore by taking a transformation which is equivalent to T-duality, the Dp-brane action is obtained. We also study type IIA/IIB NS5-brane and Kaluza-Klein monopole systems by taking other VEV assignments. Such various compactifications can be realized in the nonabelian (2,0) theory, since both longitudinal and transverse directions can be compactified, which is different from the BLG theory. We finally discuss U-duality among these branes, and show that most of the moduli parameters in U-duality group are recovered. Especially in D5-brane case, the whole U-duality relation is properly reproduced.
Planat, Michel; Saniga, Metod
2009-01-01
We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\\it Clifford group dipoles and the enactment of Weyl/Coxeter group $W(E_8)$ by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of $W(E_8)$ into the four-letter alternating group $A_4$, obtained from a chain of maximal subgroups. Group $A_4$ is realized from two B-type generators and found to correspond to the Lie algebra $sl(3,\\mathbb{C})\\oplus u(1)$. Possible applications of our findings to particle physics and the structure of genetic code are also ...
Hentosh, Oksana E.; Prykarpatsky, Yarema A.; Blackmore, Denis; Prykarpatski, Anatolij K.
2017-10-01
The work is devoted to recent investigations of the Lax-Sato compatible linear vector field equations, especially to the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebański and later analyzed in a series of articles. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equation being considered. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of the very interesting Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed.
Multi-Soft gluon limits and extended current algebras at null-infinity
McLoughlin, Tristan
2016-01-01
In this note we consider aspects of the current algebra interpretation of multi-soft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities correspond to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic c...
Meng, Qingtian; Guan, Daren; Ding, Shiliang
2001-04-01
An algebraic construction of a Hamiltonian is used to study the rotational spectra of linear triatomic molecules on the basis of the subgroup chain of symmetry U1(4)⊗ U2(4). After considering the rotation-vibration interaction which gives the l splittings, the eigenvalue expression of the Hamiltonian has a form of the term value equation commonly used in the calculation of molecular spectra. The method is applied to calculate the rotational energy levels of vibrational transitions (0 1 10-0 0 00) for C 34S 2, (1 1 13-0 1 10) and (1 0 03-0 0 00) for C 32S 2. The obtained rotational constants can represent the rotational spectra of the three bands with small root-mean-square frequency errors. The results show that the algebraic Hamiltonian can provide an alternative description of rovibrational spectra for linear triatomic molecules.
Generalized H-Lie Structure of Associative Algebras in a Category
Institute of Scientific and Technical Information of China (English)
Shuanhong Wang; Haibin Kan; Huixiang Chen
2002-01-01
We show that, if A is a sum of two H-commutative subalgebras, then the H-commutator ideal of A is nilpotent. This is inspired by a classical result of Kegel [9], which says that a ring is nilpotent if it is a sum of two nilpotent subrings. Then a partial analog of some results in [12] is shown in a more general quasi-triangular Hopf algebra setting.
Dynamics of vibrational chaos and entanglement in triatomic molecules: Lie algebraic model
Institute of Scientific and Technical Information of China (English)
Zhai Liang-Jun; Zheng Yu-Jun; Ding Shi-Liang
2012-01-01
In this paper,the dynamics of chaos and the entanglement in triatomic molecnlar vibrations are investigated.On the classical aspect,we study the chaotic trajectories in the phase space.We employ the linear entropy to examine the dynamical entanglement of the two bonds on the quantum aspect.The correspondence between the classical chaos and the quantum dynamical entanglement is also investigated.As an example,we apply our algebraic model to molecule H2O.
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
Directory of Open Access Journals (Sweden)
Vladimir S. Gerdjikov
2006-02-01
Full Text Available The construction of a family of real Hamiltonian forms (RHF for the special class of affine 1+1-dimensional Toda field theories (ATFT is reported. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. The construction method is illustrated on the explicit nontrivial example of RHF of ATFT related to the exceptional algebras E_6 and E_7. The involutions of the local integrals of motion are proved by means of the classical R-matrix approach.
A new derivation of the highest-weight polynomial of a unitary lie algebra
Energy Technology Data Exchange (ETDEWEB)
P Chau, Huu-Tai; P Van, Isacker [Grand Accelerateur National d' Ions Lourds (GANIL), 14 - Caen (France)
2000-07-01
A new method is presented to derive the expression of the highest-weight polynomial used to build the basis of an irreducible representation (IR) of the unitary algebra U(2J+1). After a brief reminder of Moshinsky's method to arrive at the set of equations defining the highest-weight polynomial of U(2J+1), an alternative derivation of the polynomial from these equations is presented. The method is less general than the one proposed by Moshinsky but has the advantage that the determinantal expression of the highest-weight polynomial is arrived at in a direct way using matrix inversions. (authors)
Trell, Erik; Edeagu, Samuel; Animalu, Alexander
2017-01-01
From a brief recapitulation of the foundational works of Marius Sophus Lie and Herrmann Günther Grassmann, and including missing African links, a rhapsodic survey is made of the straight line of extension and existence that runs as the very fibre of generation and creation throughout Nature's all utterances, which must therefore ultimately be the web of Reality itself of which the Arts and Sciences are interpreters on equal explorer terms. Assuming their direct approach, the straight line and its archaic and algebraic and artistic bearings and convolutions have been followed towards their inner reaches, which earlier resulted in a retrieval of the baryon and meson elementary particles and now equally straightforward the electron geodesics and the organic build of the periodic system of the elements.
MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS
Institute of Scientific and Technical Information of China (English)
Cui Jianlian; Choonkil Park
2012-01-01
Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B) - Φ(B)Φ(A)* =AB - BA* for all A,B ∈ A,then there exist a linear bijective map Ψ:A - A satisfying Ψ(A)Ψ(B) - Ψ(B)Ψ(A)* =AB - BA* for A,B ∈ A and a real functional h on A with h(0) =0 such that Φ(A) =Ψ(A) + h(A)I for every A ∈ A.In particular,if A is a type Ⅰ factor,then,Φ(A) =cA + h(A)I for every A ∈ A,where c =±-1.
Energy Technology Data Exchange (ETDEWEB)
Ibarra-Sierra, V.G.; Sandoval-Santana, J.C. [Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 México D.F. (Mexico); Cardoso, J.L. [Área de Física Teórica y Materia Condensada, Universidad Autónoma Metropolitana Azcapotzalco, Av. San Pablo 180, Col. Reynosa-Tamaulipas, Azcapotzalco, 02200 México D.F. (Mexico); Kunold, A., E-mail: akb@correo.azc.uam.mx [Área de Física Teórica y Materia Condensada, Universidad Autónoma Metropolitana Azcapotzalco, Av. San Pablo 180, Col. Reynosa-Tamaulipas, Azcapotzalco, 02200 México D.F. (Mexico)
2015-11-15
We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators. -- Highlights: •We deal with the general quadratic Hamiltonian and a particle in electromagnetic fields. •The evolution operator is worked out through the Lie algebraic approach. •We also obtain the propagator and Heisenberg picture position and momentum operators. •Analytical expressions for a
Uglov, D B
1997-01-01
The decompositions of the Fock and Basic modules of the affine Lie algebra constructed. Each of the irreducible submodules admits the unique up to normalization eigenbasis of the maximal commutative subalgebra of the Yangian. The elements of this eigenbasis are identified with specializations of Macdonald symmetric functions where both parameters of these functions approach an N-th primitive root of unity.
Yu, Zhang; Zhang, Yufeng
2009-01-30
Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified KdV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of solition equations.
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
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Hermann T. Tchokouansi; Victor K. Kuetche; Abbagari Souleymanou; Thomas B. Bouetou; Timoleon C. Kofane
2012-01-01
We carry out the hidden structural symmetries embedded within a system comprising ultra-short pulses which propagate in optical nonlinear media. Based upon the Wahlquist Estabrook approach, we construct the Lie-algebra valued connections associated to the previous symmetries while deriving their corresponding Lax-pairs, which are particularly useful in soliton theory. In the wake of previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + l)-dimensional ultra-short pulse equation is unveiled along with its inverse scattering formulation, the application of which are straightforward in nonlinear optics where an additional propagating dimension deserves some attention.%We carry out the hidden structural symmetries embedded within a system comprising ultra-short pulses which propagate in optical nonlinear media.Based upon the Wahlquist Estabrook approach,we construct the Liealgebra valued connections associated to the previous symmetries while deriving their corresponding Lax-pairs,which are particularly useful in soliton theory.In the wake of previous results,we extend the above prolongation scheme to higher-dimensional systems from which a new (2+ 1)-dimensional ultra-short pulse equation is unveiled along with its inverse scattering formulation,the application of which are straightforward in nonlinear optics where an additional propagating dimension deserves some attention.
Singh, S; Modi, S; Bagga, D; Kaur, P; Shankar, L R; Khushu, S
2013-03-01
The present study aimed to investigate whether brain morphological differences exist between adult hypothyroid subjects and age-matched controls using voxel-based morphometry (VBM) with diffeomorphic anatomic registration via an exponentiated lie algebra algorithm (DARTEL) approach. High-resolution structural magnetic resonance images were taken in ten healthy controls and ten hypothyroid subjects. The analysis was conducted using statistical parametric mapping. The VBM study revealed a reduction in grey matter volume in the left postcentral gyrus and cerebellum of hypothyroid subjects compared to controls. A significant reduction in white matter volume was also found in the cerebellum, right inferior and middle frontal gyrus, right precentral gyrus, right inferior occipital gyrus and right temporal gyrus of hypothyroid patients compared to healthy controls. Moreover, no meaningful cluster for greater grey or white matter volume was obtained in hypothyroid subjects compared to controls. Our study is the first VBM study of hypothyroidism in an adult population and suggests that, compared to controls, this disorder is associated with differences in brain morphology in areas corresponding to known functional deficits in attention, language, motor speed, visuospatial processing and memory in hypothyroidism. © 2012 British Society for Neuroendocrinology.
A study of the stretching vibrational spectroscopy of C120O and C120O2 by u(2 lie algebra
Directory of Open Access Journals (Sweden)
Sen Rupam
2013-01-01
Full Text Available The vibrational energy levels of endohedral fullerene dimers C120O and C120O2 are calculated considering the local Hamiltonian of Morse potential using the algebra. Here each bond of the molecules is replaced by a corresponding Lie algebra and finally the Hamiltonian is constructed considering the interacting Casimir and Majorana operators. The fundamental stretching modes of vibration of both the dimmers C120O and C120O2 are then calculated using this Hamiltonian to compare the results of functional-based tight-binding (DF-TB calculations.
Papi, Paolo; Advances in Lie Superalgebras
2014-01-01
The volume is the outcome of the conference "Lie superalgebras," which was held at the Istituto Nazionale di Alta Matematica, in 2012. The conference gathered many specialists in the subject, and the talks held provided comprehensive insights into the newest trends in research on Lie superalgebras (and related topics like vertex algebras, representation theory and supergeometry). The book contains contributions of many leading esperts in the field and provides a complete account of the newest trends in research on Lie Superalgebras.
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
Sepanski, Mark R
2010-01-01
Mark Sepanski's Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems
Another Representation for the Maximal Lie Algebra of sl(n+2,ℝ in Terms of Operators
Directory of Open Access Journals (Sweden)
Tooba Feroze
2009-01-01
of the special linear group of order (n+2, over the real numbers, sl(n+2,ℝ. In this paper, we provide an alternate representation of the symmetry algebra by simple relabelling of indices. This provides one more proof of the result that the symmetry algebra of (ya″=0 is sl(n+2,ℝ.
The Module Structure of the Extended TKK Lie Algebra%广义TKK代数的一类模结构
Institute of Scientific and Technical Information of China (English)
李鸿萍
2012-01-01
设S是欧氏空间R″（υ≥1）中最小的非格半格,在一个Jordan代数T（S）的基础上,通过Tits-Kantor-Koecher方法可构造TKK李代数g（T（S））,研究该李代数的泛中心扩张广义TKK代数g（T（S））,的一类在群代数与对称代数上的不可约表示。%Let S be the smallest possible（nonlattice） semilattice in the Euclidean space R″（υ≥1）.Form a Jordan algebra T（S）,using the Tits-Kantor-Koecher construction,we obstain TKK algebra g（T（S））.In this paper we study an Irreducible representation with group algebra and symmetric algebra states for the extended TKK algebra g（T（S））.
Energy Technology Data Exchange (ETDEWEB)
Chau, L.L.
1983-01-01
Integrable properties, i.e., existence of linear systems, infinite number of conservation laws, Reimann-Hilbert transforms, affine Lie algebra of Kac-Moody, and Bianchi-Baecklund transformation, are discussed for the constraint equations of the supersymmetric Yang-Mills fields. For N greater than or equal to 3 these constraint equations give equations of motion of the fields. These equations of motion reduce to the ordinary Yang-Mills equations as the spinor and scalar fields are eliminated. These understandings provide a possible method to solve the full Yang-Mills equations. Connections with other non-linear systems are also discussed. 53 references.
Quasitriangular chiral WZW model in a nutshell
Klimcík, C
2001-01-01
We give the bare-bone description of the quasitriangular chiral WZW model for the particular choice of the Lu-Weinstein-Soibelman Drinfeld double of the affine Kac-Moody group. The symplectic structure of the model and its Poisson-Lie symmetry are completely characterized by two $r$-matrices with spectral parameter. One of them is ordinary and trigonometric and characterizes the $q$-current algebra. The other is dynamical and elliptic (in fact Felder's one) and characterizes the braiding of $q$-primary fields.
Dotsenko, V.; Shadrin, S.; Vallette, B.
2016-01-01
In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for preLie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated
24 faces of the Borcherds modular form Phi_{12}
Gritsenko, Valery
2012-01-01
The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebras, Siegel modular forms and affine Lie algebras.
Energy Technology Data Exchange (ETDEWEB)
Campoamor-Stursberg, R [Dpto. GeometrIa y TopologIa Fac. CC. Matematicas Universidad Complutense de Madrid Plaza de Ciencias, 3 E-28040 Madrid (Spain)], E-mail: rutwig@pdi.ucm.es
2008-08-15
We show that the Inoenue-Wigner contraction naturally associated to a reduction chain s implies s' of semisimple Lie algebras induces a decomposition of the Casimir operators into homogeneous polynomials, the terms of which can be used to obtain additional mutually commuting missing label operators for this reduction. The adjunction of these scalars that are no more invariants of the contraction allow to solve the missing label problem for those reductions where the contraction provides an insufficient number of labelling operators.
Sugawara construction for higher genus Riemann surfaces
Schlichenmaier, Martin
1999-04-01
By the classical genus zero Sugawara construction one obtains representations of the Virasoro algebra from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type). In this lecture, the classical construction is recalled first. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in a joint work with O. K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains (from representations of the global algebras of affine type) representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is a generalization of the Virasoro algebra to higher genus.
Lie Algebraic Structures and Integrability of Long-Short Wave Equation in （2＋1）Dimensions
Institute of Scientific and Technical Information of China (English)
ZHAOXue-Qing; LüJing-Fa
2004-01-01
The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.
D p-branes, NS5-branes and U-duality from nonabelian (2,0) theory with Lie 3-algebra
Honma, Yoshinori; Ogawa, Morirou; Shiba, Shotaro
2011-04-01
We derive the super Yang-Mills action of D p-branes on a torus T p-4 from the nonabelian (2, 0) theory with Lie 3-algebra [1]. Our realization is based on Lie 3-algebra with pairs of Lorentzian metric generators. The resultant theory then has negative norm modes, but it results in a unitary theory by setting VEV's of these modes. This procedure corresponds to the torus compactification, therefore by taking a transformation which is equivalent to T-duality, the D p-brane action is obtained. We also study type IIA/IIB NS5brane and Kaluza-Klein monopole systems by taking other VEV assignments. Such various compactifications can be realized in the nonabelian (2, 0) theory, since both longitudinal and transverse directions can be compactified, which is different from the BLG theory. We finally discuss U-duality among these branes, and show that most of the moduli parameters in U-duality group are recovered. Especially in D5-brane case, the whole U-duality relation is properly reproduced.
A Kind of Braided-Lie Structures
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
@@ We introduce a family of braidedLie algebras.They are Lie algebras in the unifying YetterDrinfeldLong module categoryJJMQQ where J and Q are Hopf algebras.We study their structure and the braidedLie structure of an algebra A in JJM QQ.
Institute of Scientific and Technical Information of China (English)
An Hui-hui; Wang Zhi-chun
2016-01-01
L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.
Iachello, F
1995-01-01
1. The Wave Mechanics of Diatomic Molecules. 2. Summary of Elements of Algebraic Theory. 3. Mechanics of Molecules. 4. Three-Body Algebraic Theory. 5. Four-Body Algebraic Theory. 6. Classical Limit and Coordinate Representation. 8. Prologue to the Future. Appendices. Properties of Lie Algebras; Coupling of Algebras; Hamiltonian Parameters
Central simple Poisson algebras
Institute of Scientific and Technical Information of China (English)
SU Yucai; XU Xiaoping
2004-01-01
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
Brown, Ronald E
2007-01-01
We define the notion of whiskered categories and groupoids and discuss potential applications, relations betweens topics, extensions, for example to a many object Lie theory, to automorphism structures for crossed modules, and to resolutions of monoids. This paper is more an outline of a possible programme or programmes and their relationships than giving conclusive results.
Stochastic Lie group integrators
Malham, Simon J A
2007-01-01
We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the Lie group and then to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. These involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They become stochastic Lie group integrator schemes if...
Institute of Scientific and Technical Information of China (English)
Daxing Wu; Ying Zhao; Jian Liao; Huifang Yin; Wei Wang
2011-01-01
Voxel-based morphometry-diffeomorphic anatomical registration using exponentiated lie algebra analysis was used to investigate the structural characteristics of white matter in young males with antisocial personality disorder (APD) and healthy controls without APD. The results revealed that APD subjects, relative to healthy subjects, exhibited increased white matter volume in the bilateral prefrontal lobe, right insula, precentral gyrus, bilateral superior temporal gyrus, right postcentral gyrus, right inferior parietal lobule, right precuneus, right middle occipital lobe, right parahippocampal gyrus and bilateral cingulate, and decreased volume in the middle temporal cortex and right cerebellum. The white matter volume in the medial frontal gyrus was significantly correlated with antisocial type scores on the Personality Diagnostic Questionnaire in APD subjects. These experimental findings indicate that white matter abnormalities in several brain areas may contribute to antisocial behaviors in APD subjects.
Das, Tapas
2015-01-01
The second order $N$-dimensional Schr\\"odinger equation with pseudoharmonic potential is reduced to a first order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Our results generalize all other previous works that done for various potential combinations in the case of lower dimensions.The Ladder operators are also constructed for the pseudoharmonic potential in $N$-dimensions.Lie algebra associated with these operators are studied and found that they satisfy the commutation relations for the SU(1,1) group. Matrix elements of different operators such as $z$, $z\\frac{d}{dz}$ are derived and finally the Casimir operator is discussed briefly.
Yangians and transvector algebras
Molev, A. I.
1998-01-01
Olshanski's centralizer construction provides a realization of the Yangian for the Lie algebra gl(n) as a subalgebra in the projective limit of a chain of centralizers in the universal enveloping algebras. We give a modified version of this construction based on a quantum analog of Sylvester's theorem. We then use it to get an algebra homomorphism from the Yangian to the transvector algebra associated with the general linear Lie algebras. The results are applied to identify the elementary rep...
Isomorphism of Intransitive Linear Lie Equations
Directory of Open Access Journals (Sweden)
Jose Miguel Martins Veloso
2009-11-01
Full Text Available We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.
Lie bialgebras of generalized Witt type
Institute of Scientific and Technical Information of China (English)
SONG; Guang'ai; SU; Yucai
2006-01-01
In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W (x) W) is trivial.
Quiver representations and quiver varieties
Jr, Alexander Kirillov
2016-01-01
This book is an introduction to the theory of quiver representations and quiver varieties, starting with basic definitions and ending with Nakajima's work on quiver varieties and the geometric realization of Kac-Moody Lie algebras. The first part of the book is devoted to the classical theory of quivers of finite type. Here the exposition is mostly self-contained and all important proofs are presented in detail. The second part contains the more recent topics of quiver theory that are related to quivers of infinite type: Coxeter functor, tame and wild quivers, McKay correspondence, and representations of Euclidean quivers. In the third part, topics related to geometric aspects of quiver theory are discussed, such as quiver varieties, Hilbert schemes, and the geometric realization of Kac-Moody algebras. Here some of the more technical proofs are omitted; instead only the statements and some ideas of the proofs are given, and the reader is referred to original papers for details. The exposition in the book requ...
Algebra Automorphisms of Quantized Enveloping Algebras Uq(■)
Institute of Scientific and Technical Information of China (English)
查建国
1994-01-01
The algebra automorphisms of the quantized enveloping algebra Uq(g) are discussed, where q is generic. To some extent, all quantum deformations of automorphisms of the simple Lie algebra g have been determined.
Energy Technology Data Exchange (ETDEWEB)
Goto, Masami; Ino, Kenji; Yano, Keiichi [University of Tokyo Hospital, Department of Radiological Technology, Bunkyo-ku, Tokyo (Japan); Abe, Osamu [Nihon University School of Medicine, Department of Radiology, Itabashi-ku, Tokyo (Japan); Aoki, Shigeki [Juntendo University, Department of Radiology, Bunkyo-ku, Tokyo (Japan); Hayashi, Naoto [University of Tokyo Hospital, Department of Computational Diagnostic Radiology and Preventive Medicine, Bunkyo-ku, Tokyo (Japan); Miyati, Tosiaki [Kanazawa University, Graduate School of Medical Science, Kanazawa (Japan); Takao, Hidemasa; Mori, Harushi; Kunimatsu, Akira; Ohtomo, Kuni [University of Tokyo Hospital, Department of Radiology and Department of Computational Diagnostic Radiology and Preventive Medicine, Bunkyo-ku, Tokyo (Japan); Iwatsubo, Takeshi [University of Tokyo, Department of Neuropathology, Bunkyo-ku, Tokyo (Japan); Yamashita, Fumio [Iwate Medical University, Department of Radiology, Yahaba, Iwate (Japan); Matsuda, Hiroshi [Integrative Brain Imaging Center National Center of Neurology and Psychiatry, Department of Nuclear Medicine, Kodaira, Tokyo (Japan); Collaboration: Japanese Alzheimer' s Disease Neuroimaging Initiative
2013-07-15
This study aimed to investigate whether the effect of scanner for cortex volumetry with atlas-based method is reduced using Diffeomorphic Anatomical Registration Through Exponentiated Lie Algebra (DARTEL) normalization compared with standard normalization. Three-dimensional T1-weighted magnetic resonance images (3D-T1WIs) of 21 healthy subjects were obtained and evaluated for effect of scanner in cortex volumetry. 3D-T1WIs of the 21 subjects were obtained with five MRI systems. Imaging of each subject was performed on each of five different MRI scanners. We used the Voxel-Based Morphometry 8 tool implemented in Statistical Parametric Mapping 8 and WFU PickAtlas software (Talairach brain atlas theory). The following software default settings were used as bilateral region-of-interest labels: ''Frontal Lobe,'' ''Hippocampus,'' ''Occipital Lobe,'' ''Orbital Gyrus,'' ''Parietal Lobe,'' ''Putamen,'' and ''Temporal Lobe.'' Effect of scanner for cortex volumetry using the atlas-based method was reduced with DARTEL normalization compared with standard normalization in Frontal Lobe, Occipital Lobe, Orbital Gyrus, Putamen, and Temporal Lobe; was the same in Hippocampus and Parietal Lobe; and showed no increase with DARTEL normalization for any region of interest (ROI). DARTEL normalization reduces the effect of scanner, which is a major problem in multicenter studies. (orig.)
Goto, Masami; Abe, Osamu; Aoki, Shigeki; Hayashi, Naoto; Miyati, Tosiaki; Takao, Hidemasa; Iwatsubo, Takeshi; Yamashita, Fumio; Matsuda, Hiroshi; Mori, Harushi; Kunimatsu, Akira; Ino, Kenji; Yano, Keiichi; Ohtomo, Kuni
2013-07-01
This study aimed to investigate whether the effect of scanner for cortex volumetry with atlas-based method is reduced using Diffeomorphic Anatomical Registration Through Exponentiated Lie Algebra (DARTEL) normalization compared with standard normalization. Three-dimensional T1-weighted magnetic resonance images (3D-T1WIs) of 21 healthy subjects were obtained and evaluated for effect of scanner in cortex volumetry. 3D-T1WIs of the 21 subjects were obtained with five MRI systems. Imaging of each subject was performed on each of five different MRI scanners. We used the Voxel-Based Morphometry 8 tool implemented in Statistical Parametric Mapping 8 and WFU PickAtlas software (Talairach brain atlas theory). The following software default settings were used as bilateral region-of-interest labels: "Frontal Lobe," "Hippocampus," "Occipital Lobe," "Orbital Gyrus," "Parietal Lobe," "Putamen," and "Temporal Lobe." Effect of scanner for cortex volumetry using the atlas-based method was reduced with DARTEL normalization compared with standard normalization in Frontal Lobe, Occipital Lobe, Orbital Gyrus, Putamen, and Temporal Lobe; was the same in Hippocampus and Parietal Lobe; and showed no increase with DARTEL normalization for any region of interest (ROI). DARTEL normalization reduces the effect of scanner, which is a major problem in multicenter studies.
交换环上一些线性李代数的导子%Derivations of Certain Linear Lie Algebras over Commutative Rings
Institute of Scientific and Technical Information of China (English)
偶世坤; 王登银; 夏春光
2009-01-01
Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this paper, we first determine the intermediate algebras between h and b, then for such an intermediate algebra, we give an explicit description on its derivations, provided that R is a commutative ring with identity and 2 is invertible in R.
Institute of Scientific and Technical Information of China (English)
Gao Ya-Jun
2004-01-01
By proposing a so-called extended hyperbolic complex (EHC) function method, an Ernst-like (p + 2) × (p + 2)matrix EHC potential is introduced for the stationary axisymmetric (SAS) Einstein-Maxwell theory with p Abelian gauge fields (EM-p theory, for short), then the field equations of the SAS EM-p theory are written as a so-called HauserErnst-like self-dual relation for the EHC matrix potential. Two Hauser-Ernst-type EHC linear systems are established,based on which some new parametrized symmetry transformations for the SAS EM-p theory are explicitly constructed.These hidden symmetries are found to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac-Moody algebra su(p + 1, 1) R(t, t-1) and Virasoro algebra (without centre charges). All of the SAS EM-p gravity and the Einstein-Maxwell cases.
El-Chaar, Caroline
2012-01-01
In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. Using this fourth realization, we explicitly describe all its ideals.
Lie groups and automorphic forms
Ji, Lizhen; Xu, H W; Yau, Shing-Tung
2006-01-01
Lie groups are fundamental objects in mathematics. They occur naturally in differential geometry, algebraic geometry, representation theory, number theory, and other areas. Closely related are arithmetic subgroups, locally symmetric spaces and the spectral theory of automorphic forms. This book consists of five chapters which give comprehensive introductions to Lie groups, Lie algebras, arithmetic groups and reduction theories, cohomology of arithmetic groups, and the Petersson and Kuznetsov trace formulas.
Indian Academy of Sciences (India)
Antonio J Calderón Martín; Manuel Forero Piulestán; José M Sánchez Delgado
2012-05-01
We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras is of the form $M=\\mathcal{U}+\\sum_jI_j$ with $\\mathcal{U}$ a subspace of the abelian Malcev subalgebra and any $I_j$ a well described ideal of satisfying $[I_j, I_k]=0$ if ≠ . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.
Evolution algebras and their applications
Tian, Jianjun Paul
2008-01-01
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
Krichever, Igor M.; Sheinman, Oleg K.
2007-01-01
In this paper we develop a general concept of Lax operators on algebraic curves introduced in [1]. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct the orthogonal and symplectic analogs of Lax operators, prove that they constitute almost graded Lie algebras and construct local central extensions of those Lie algebras.
Madjumdar-Papapetrou Type Solutions in Sigma-model and Intersecting p-branes
Ivashchuk, V D
1999-01-01
The block-orthogonal generalization of the Madjumdar-Papapetrou type solutions for the sigma-model studied earlier in \\cite{IM4}-\\cite{IMC} are obtained and corresponding solutions with p-branes are considered. The existence of solutions and the number of independent harmonic functions is defined by the matrix of scalar products of vectors $U^s$, governing the sigma-model target space metric. (For orthogonal $U^s$, when target space is symmetric homogeneous space, the solutions coincide with those from finite dimensional Lie algebras and hyperbolic (Kac-Moody) algebras are singled out and investigated. The affine Cartan matrices do not arise in the scheme under consideration. Some examples of solutions and intersection rules for D=11 supergravity, related D=12 theory and extending them $B_D$-models are considered. For special multicenter solutions the indicators of horizon and curvature singularity are introduced.
Celeghini, Enrico
2014-01-01
Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A Rigged Hilbert space is found and a new discrete basis in R obtained. The operators {O[R]} defined on R are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in this way, their algebraic discussion. Introducing in the half-line a Fourier-like Transform, the procedure is extended to R^+ and can be easily generalized to R^n and to spherical reference systems.
Noncommutative Poisson brackets on Loday algebras and related deformation quantization
UCHINO, Kyousuke
2010-01-01
We introduce a new type of algebra which is called a Loday-Poisson algebra. The class of the Loday-Poisson algebras forms a special subclass of Aguiar's dual-prePoisson algebas (\\cite{A}). We will prove that there exists a unique Loday-Poisson algebra over a Loday algebra, like the Lie-Poisson algebra over a Lie algebra. Thus, Loday-Poisson algebras are regarded as noncommutative analogues of Lie-Poisson algebras. We will show that the polinomial Loday-Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday's associative dialgebra.
Construction of complete generalized algebraic groups
Institute of Scientific and Technical Information of China (English)
WANG; Dengyin
2005-01-01
With one exception, the holomorph of a finite dimensional abelian connectedalgebraic group is shown to be a complete generalized algebraic group. This result on algebraic group is an analogy to that on Lie algebra.
Institute of Scientific and Technical Information of China (English)
张萍; 高寿兰; 孙天川
2015-01-01
Many kinds of generations and deformations of Schrödinger Virasoro algebra have been studied recently. In this paper,we calculate all the Leibniz 2 cocycles of 2 dimensional central extensions of a type of twisted de-formative Schrödinger Virasoro Lie algebra.Thus its Leibniz central extension is determined.%近来各种 Schrödinger Virasoro 李代数推广与变形得到了广泛的研究。本文计算一类Schrödinger Virasoro 李代数2维中心扩张所有的 Leibniz 2上循环，从而确定了这类李代数的 Leibniz中心扩张。
Generalized derivations of Lie triple systems
Directory of Open Access Journals (Sweden)
Zhou Jia
2016-01-01
Full Text Available In this paper, we present some basic properties concerning the derivation algebra Der (T, the quasiderivation algebra QDer (T and the generalized derivation algebra GDer (T of a Lie triple system T, with the relationship Der (T ⊆ QDer (T ⊆ GDer (T ⊆ End (T. Furthermore, we completely determine those Lie triple systems T with condition QDer (T = End (T. We also show that the quasiderivations of T can be embedded as derivations in a larger Lie triple system.
Classifying Two-dimensional Hyporeductive Triple Algebras
Issa, A Nourou
2010-01-01
Two-dimensional real hyporeductive triple algebras (h.t.a.) are investigated. A classification of such algebras is presented. As a consequence, a classification of two-dimensional real Lie triple algebras (i.e. generalized Lie triple systems) and two-dimensional real Bol algebras is given.
The Virasoro vertex algebra and factorization algebras on Riemann surfaces
Williams, Brian
2017-08-01
This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.
Institute of Scientific and Technical Information of China (English)
SU; Yucai(
2001-01-01
［1］ Kawamoto, N., Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J., 1986, 16: 417.［2］ Osborn, J. M., New simple infinite－dimensional Lie algebras of characteristic 0, J. Alg., 1996, 185: 820.［3］ Dokovic, D. Z., Zhao, K., Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Trans. of Amer. Math. Soc., 1998, 350(2): 643.［4］ Dokovic, D. Z., Zhao, K., Generalized Cartan type W Lie algebras in characteristic zero, J. Alg., 1997, 195: 170.［5］ Osborn, J. M., Zhao, K., Generalized Poisson bracket and Lie algebras of type H in characteristic 0, Math. Z., 1999, 230: 107.［6］ Osborn, J. M., Zhao, K., Generalized Cartan type K Lie algebras in characteristic 0, Comm. Alg., 1997, 25: 3325.［7］ Zhao, K., Isomorphisms between generalized Cartan type W Lie algebras in characteristic zero, Canadian J. Math., 1998, 50: 210.［8］ Passman, D. P., Simple Lie algebras of Witt type, J. Algebra, 1998, 206: 682.［9］ Jordan, D. A., On the simplicity of Lie algebras of derivations of commutative algebras, J. Alg., 2000, 206: 682.［10］ Xu, X., New generalized simple Lie algebras of Cartan type over a field with characteristic 0, J. Alg., 2000, 244: 23.［11］ Su, Y., Xu, X., Zhang, H., Derivation－simple algebras and the structures of Lie algebras of generalized Witt type, J. Alg., 2000, 233: 642.［12］ Dixmer, J., Enveloping Algebras, Amsterdam: North Holland, 1977.
Derivations of generalized Weyl algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai(苏育才)
2003-01-01
A class of the associative and Lie algebras A[D] = A × F[D] of Weyl type are studied, where Ais a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] isthe polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A suchthat A is D-simple. The derivations of these associative and Lie algebras are precisely determined.
String Topology for Lie Groups
DEFF Research Database (Denmark)
A. Hepworth, Richard
2010-01-01
In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer ...
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Over a field F of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector space A［D]=A［D] from any pair of a commutative associative algebra A with an identity element and the polynomial algebra ［D] of a commutative derivation subalgebra D of A. We prove that A[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only if A is D－simple and A［D] acts faithfully on A. Thus we obtain a lot of simple algebras.
Simple Algebras of Invariant Operators
Institute of Scientific and Technical Information of China (English)
Xiaorong Shen; J.D.H. Smith
2001-01-01
Comtrans algebras were introduced in as algebras with two trilinear operators, a commutator [x, y, z] and a translator , which satisfy certain identities. Previously known simple comtrans algebras arise from rectangular matrices, simple Lie algebras, spaces equipped with a bilinear form having trivial radical, spaces of hermitian operators over a field with a minimum polynomial x2+1. This paper is about generalizing the hermitian case to the so-called invariant case. The main result of this paper shows that the vector space of n-dimensional invariant operators furnishes some comtrans algebra structures, which are simple provided that certain Jordan and Lie algebras are simple.
Central extensions of Lax operator algebras
Schlichenmaier, M.; Sheinman, O. K.
2008-08-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.
Knippenberg, Stefan; Starcke, Jan-Hendrik; Wormit, Michael; Dreuw, Andreas
2010-01-01
Abstract The vertical excited electronic states of linearly fused neutral polyacenes and their radical cations have been investigated using the algebraic diagrammatic construction scheme of sec- ond order (ADC(2)). While strict ADC(2) (ADC(2)-s) correctly reproduces trends for mainly singly excited states, in extended ADC(2) (ADC(2)-x) the description of doubly excited states is critically improved. It is shown that a combined application of strict and extended ADC(2) nicely reveal...
An introduction to algebraic geometry and algebraic groups
Geck, Meinolf
2003-01-01
An accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups of Lie type.The text covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups
Dijkgraaf, R
1998-01-01
Fivebranes are non-perturbative objects in string theory that generalize two-dimensional conformal field theory and relate such diverse subjects as moduli spaces of vector bundles on surfaces, automorphic forms, elliptic genera, the geometry of Calabi-Yau threefolds, and generalized Kac-Moody algebras.
Xi-Zhong, Liu
2012-01-01
In this paper, We derive the symmetry group theorem to the Lin-Tsien equation by using the modified CK's direct method, from which we obtain the corresponding symmetry group. More importantly, conservation laws corresponding to the Kac-Moody-Virasoro symmetry algebra of Lin-Tsien equation is obtained up to second order group invariants.
Maximal subalgebras of the special linear Lie algebras over commutative ring%交换环上特殊线性李代数的极大子代数
Institute of Scientific and Technical Information of China (English)
刘洋; 刘文德
2016-01-01
In this paper, we determine all maximal subalgebras of the special linear Lie algebra containing the canonical torus using maximal ideas over a unital commutative ring with 2, 3 be the unit. We also determine the number of maximal subslgebras and prove that each maximal subslgebra is conjugate under a permutation matrix to a standard one.%文章利用有单位元且2,3是单位的交换环的极大理想刻画了其上特殊线性李代数包含典范环面的极大子代数。确定了特殊线性李代数极大子代数的个数,并证明了每个极大子代数均可通过置换矩阵共轭于标准的极大子代数。
Twisted derivations of Hopf algebras
Davydov, Alexei
2012-01-01
In the paper we introduce the notion of twisted derivation of a bialgebra. Twisted derivations appear as infinitesimal symmetries of the category of representations. More precisely they are infinitesimal versions of twisted automorphisms of bialgebras. Twisted derivations naturally form a Lie algebra (the tangent algebra of the group of twisted automorphisms). Moreover this Lie algebra fits into a crossed module (tangent to the crossed module of twisted automorphisms). Here we calculate this crossed module for universal enveloping algebras and for the Sweedler's Hopf algebra.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
Structure and representations of Jordan algebras
Jacobson, Nathan
1968-01-01
The theory of Jordan algebras has played important roles behind the scenes of several areas of mathematics. Jacobson's book has long been the definitive treatment of the subject. It covers foundational material, structure theory, and representation theory for Jordan algebras. Of course, there are immediate connections with Lie algebras, which Jacobson details in Chapter 8. Of particular continuing interest is the discussion of exceptional Jordan algebras, which serve to explain the exceptional Lie algebras and Lie groups. Jordan algebras originally arose in the attempts by Jordan, von Neumann,
The non-commutative Weil algebra
1999-01-01
Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism fo...
Superderivations for a Family of Lie Superalgebras of Special Type*
Institute of Scientific and Technical Information of China (English)
SUN XIU-MEI; ZOU XU-JUAN; LIU WEN-DE
2011-01-01
By means of generators, superderivations are completely determined for a family of Lie superalgebras of special type, the tensor products of the exterior algebras and the finite-dimensional special Lie algebras over a field of characteristic p ＞ 3. In particular, the structure of the outer superderivation algebra is concretely formulated and the dimension of the first cohomology group is given.
3-Leibniz bialgebras (3-Lie bialgebras)
2016-01-01
In this paper by use of cohomology complex of $3$-Leibniz algebras, the definitions of Leibniz bialgebras (and Lie bialgebras) are extended for the case of $3$-Leibniz algebras. Many theorems about Leibniz bialgebras are extended and proved for the case of $3$-Leibniz bialgebras ($3$-Lie bialgebras). Moreover a new theorem on the correspondence between $3$-Leibniz bialgebra and its associated Leibniz bialgebra is proved. $3$-Lie bialgebra as particular case of the $3$-Leibniz bialgebra is inv...
Semi-Hopf Algebra and Supersymmetry
Gunara, Bobby Eka
1999-01-01
We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N=1 and N=2 supersymmetry.
Prolongation structures for supersymmetric equations
Roelofs, G.H.M.; Hijligenberg, van den N.W.
1990-01-01
The well known prolongation technique of Wahlquist and Estabrook (1975) for nonlinear evolution equations is generalized for supersymmetric equations and applied to the supersymmetric extension of the KdV equation of Manin-Radul. Using the theory of Kac-Moody Lie superalgebras, the explicit form of
Universal Algebra Applied to Hom-Associative Algebras, and More
Hellström, Lars; Makhlouf, Abdenacer; Silvestrov, Sergei D.
2014-01-01
The purpose of this paper is to discuss the universal algebra theory of hom-algebras. This kind of algebra involves a linear map which twists the usual identities. We focus on hom-associative algebras and hom-Lie algebras for which we review the main results. We discuss the envelopment problem, operads, and the Diamond Lemma; the usual tools have to be adapted to this new situation. Moreover we study Hilbert series for the hom-associative operad and free algebra, and describe them up to total...
Rota-Baxter operators on Witt and Virasoro algebras
Gao, Xu; Liu, Ming; Bai, Chengming; Jing, Naihuan
2016-10-01
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
Lie symmetries and differential galois groups of linear equations
Oudshoorn, W.R.; Put, M. van der
2002-01-01
For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In co
Bosonization and Lie Group Structure
Ha, Yuan K
2015-01-01
We introduce a concise quantum operator formula for bosonization in which the Lie group structure appears in a natural way. The connection between fermions and bosons is found to be exactly the connection between Lie group elements and the group parameters. Bosonization is an extraordinary way of expressing the equation of motion of a complex fermion field in terms of a real scalar boson in two dimensions. All the properties of the fermion field theory are known to be preserved under this remarkable transformation with substantial simplification and elucidation of the original theory, much like Lie groups can be studied by their Lie algebras.
Institute of Scientific and Technical Information of China (English)
陈琼; 陈正新
2011-01-01
设F是特征为零的域,gl(n,F)为域F上的一般线性李代数,Tn为域F上全体n×n阶上三角矩阵李代数,称gl(n,F)中包含Tn的所有子代数为gl(n,F)的抛物子代数.决定出gl(n,F)上的任意标准抛物子代数P的形式,证明了任意抛物子代数P上的映射φ是保李积的非线性可逆映射当且仅当存在可逆矩阵T∈P,映射x:P→F和域F的自同构f,使得φ([aij])=T[f(aij)]T-1+x([aij])I或φ([aij])=-R(T[f(aij)]T-1)tR-1+x([atj])I,对任意的[aij]∈P,其中R=n∑i=1(-1)iE1.n+1-i,x满足对任意的A∈(-P)={[x,y]|x,y∈P},总有x(A)=0.%Let F be an arbitrary field with characteristics zero, gl(n ,F) the general linear Lie algebra of all n × n matrices, and let Tn be the Lie algebra of all upper n × n matrices. A subalgebra P of gl(n ,F) containing Tn is called a parabolic subalgebra of gl(n,F) . Decide the form of arbitrary parabolic subalgebra P of gl(n,F) and prove that a non-linear bijective map Φ on P preserves Lie products if and only if there exist an invertible matrix T∈ P , a function X:P → F satisying X(A)= 0 for every matrix A∈ P= {[x,y]1x,y ∈ P} , and an automorphism f of the field F, such that Φ([aij]) = T[f(aij)]T-1 + X([aij])I, or Φ([aij]) =- R(T[f(aij)]T-1)tR-1 + X([aij])I ,for all [aij]∈ P,where R= nΣi=1 (- 1)iE1,n+1-i .
Temme, F P
2004-03-01
The physics of dual group scalar invariants (SIs) as (Lie algebraic) group measures (L-GMs) and its significance to non-Abelian NMR spin systems motivates this overview of uniform general-2n [AX](2n) spin evolution, which represents an extensive addendum to Corio's earlier (essentially restricted) view of Abelian spin system SU(2)-based SI-cardinalities. The [Formula: see text] values in [J. Magn. Reson., 134 (1998) 131] arise from strictly linear recoupled time-reversal invariance (TRI) models. In contrast, here we discuss the physical significance of an alternative polyhedral combinatorics approach to democratic recoupling (DR), a property inherent in both the TRI and statistical sampling. Recognition of spin ensemble SIs as being L-GMs over isomorphic algebras is invaluable in many DR-based NMR problems. Various [AX]n model spin systems, including the [AX]3 bis odd-odd parity spin system, are examined as direct applications of these L-GM- and combinatorial-based SI ideas. Hence in place of /SI/=15 (implied by Corio's [Formula: see text] approach), the bis 3-fold spin system cardinality is seen now as constrained to a single invariant on an isomorphic product algebra under L-GMs, in accord with the subspectral analysis of Jones et al. [Canad. J. Chem., 43 (1965) 683]. The group projective ideas cited here for DR (as cf. to graph theoretic views) apply to highly degenerate non-Abelian problems. Over dual tensorial bases, they define models of spin dynamical evolution whose (SR) quasiparticle superboson carrier (sub)spaces are characterised by SIs acting as explicit auxiliary labels [Physica, A198 (1993) 245; J. Math. Chem., 31 (2002) 281]. A deeper [Formula: see text] network-based view of spin-alone space developed in Balasubramanian's work [J. Chem. Phys., 78 (1983) 6358] is especially important, (e.g.) in the study of spin waves [J. Math. Chem., 31 (2002) 363]. Beyond the specific NMR SIs derived here, there are DR applications where a sporadic, still higher, 2
DERIVATIONS ON DIFFERENTIAL OPERATOR ALGEBRA AND WEYL ALGEBRA
Institute of Scientific and Technical Information of China (English)
CHENCAOYU
1996-01-01
Let L be an n-dimensional nilpotent Lie algebra with a basis{x1…,xn),and every xi acts as a locally nilpotent derivation on algebra A. This paper shows that there exists a set of derivations{y1,…,yn}on U(L) such that (A#U(L))#k{y,1,…,yn] is ismorphic to the Weyl algebra An(A).The author also uses the de4rivations to obtain a necessary and sufficient condition for a finite dimesional Lie algebra to be nilpotent.
Shafarevich, I
1994-01-01
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
The Structures for the Loop-Witt Algebra
Institute of Scientific and Technical Information of China (English)
Xiao Min TANG; Zhuo ZHANG
2012-01-01
The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra.In this paper we study the universal central extension,derivations and automorphism group for the loop-Witt algebra.
Energy Technology Data Exchange (ETDEWEB)
Yoshiura, Takashi; Hiwatashi, Akio; Yamashita, Koji; Takayama, Yukihisa; Kamano, Norihiro; Honda, Hiroshi [Kyushu University, Department of Clinical Radiology, Graduate School of Medical Sciences, 3-1-1 Maidashi, Higashi-ku, Fukuoka (Japan); Ohyagi, Yasumasa; Kira, Jun-ichi [Kyushu University, Department of Neurology, Graduate School of Medical Sciences, 3-1-1 Maidashi, Higashi-ku, Fukuoka (Japan); Monji, Akira; Kawashima, Toshiro [Kyushu University, Department of Neuropsychiatry, Graduate School of Medical Sciences, 3-1-1 Maidashi, Higashi-ku, Fukuoka (Japan)
2011-02-15
To determine which brain regions are relevant to deterioration in abstract reasoning as measured by Raven's Colored Progressive Matrices (CPM) in the context of dementia. MR images of 37 consecutive patients including 19 with Alzheimer's disease (AD) and 18 with amnestic mild cognitive impairment (aMCI) were retrospectively analyzed. All patients were administered the CPM. Regional grey matter (GM) volume was evaluated according to the regimens of voxel-based morphometry, during which a non-linear registration algorithm called Diffeomorphic Anatomical Registration Through Exponentiated Lie algebra was employed. Multiple regression analyses were used to map the regions where GM volumes were correlated with CPM scores. The strongest correlation with CPM scores was seen in the left middle frontal gyrus while a region with the largest volume was identified in the left superior temporal gyrus. Significant correlations were seen in 14 additional regions in the bilateral cerebral hemispheres and right cerebellum. Deterioration of abstract reasoning ability in AD and aMCI measured by CPM is related to GM loss in multiple regions, which is in close agreement with the results of previous activation studies. (orig.)
Yoshiura, Takashi; Hiwatashi, Akio; Yamashita, Koji; Ohyagi, Yasumasa; Monji, Akira; Takayama, Yukihisa; Kamano, Norihiro; Kawashima, Toshiro; Kira, Jun-Ichi; Honda, Hiroshi
2011-02-01
To determine which brain regions are relevant to deterioration in abstract reasoning as measured by Raven's Colored Progressive Matrices (CPM) in the context of dementia. MR images of 37 consecutive patients including 19 with Alzheimer's disease (AD) and 18 with amnestic mild cognitive impairment (aMCI) were retrospectively analyzed. All patients were administered the CPM. Regional grey matter (GM) volume was evaluated according to the regimens of voxel-based morphometry, during which a non-linear registration algorithm called Diffeomorphic Anatomical Registration Through Exponentiated Lie algebra was employed. Multiple regression analyses were used to map the regions where GM volumes were correlated with CPM scores. The strongest correlation with CPM scores was seen in the left middle frontal gyrus while a region with the largest volume was identified in the left superior temporal gyrus. Significant correlations were seen in 14 additional regions in the bilateral cerebral hemispheres and right cerebellum. Deterioration of abstract reasoning ability in AD and aMCI measured by CPM is related to GM loss in multiple regions, which is in close agreement with the results of previous activation studies.
Quantum cluster algebra structures on quantum nilpotent algebras
Goodearl, K R
2017-01-01
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts.
Quasi-big\\`ebres de Lie et cohomologie d'alg\\`ebre de Lie
Bangoura, Momo
2010-01-01
Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional (G, \\mu, \\gamma ,\\phi ?), correspond one Lie algebra structure on D = G\\oplus G*, called the double of the given Lie quasi-bialgebra. We show that there exist on \\Lambda G, the exterior algebra of G, a D-module structure and we establish an isomorphism of D-modules between \\Lambda D and End(\\Lambda G), D acting on \\Lambda D by the adjoint action.
Left Artinian Algebraic Algebras
Institute of Scientific and Technical Information of China (English)
S. Akbari; M. Arian-Nejad
2001-01-01
Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.
Algebraic partial Boolean algebras
Energy Technology Data Exchange (ETDEWEB)
Smith, Derek [Math Department, Lafayette College, Easton, PA 18042 (United States)
2003-04-04
Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A{sub 5} sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E{sub 8}.
Lie Superalgebras arising from bosonic representation
Jing, Naihuan
2012-01-01
A 2-toroidal Lie superalgebra is constructed using bosonic fields and a ghost field. The superalgebra contains $osp(1|2n)^{(1)}$ as a distinguished subalgebra and behaves similarly to the toroidal Lie superalgebra of type $B(0, n)$. Furthermore this algebra is a central extension of the algebra $osp(1|2n)\\otimes \\mathbb C[s, s^{-1}, t,t^{-1}]$.
Noncommutative algebra and geometry
De Concini, Corrado; Vavilov, Nikolai 0
2005-01-01
Finite Galois Stable Subgroups of Gln. Derived Categories for Nodal Rings and Projective Configurations. Crowns in Profinite Groups and Applications. The Galois Structure of Ambiguous Ideals in Cyclic Extensions of Degree 8. An Introduction to Noncommutative Deformations of Modules. Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II. Quotient Grothendieck Representations. On the Strong Rigidity of Solvable Lie Algebras. The Role of Bergman in Invesigating Identities in Matrix Algebras with Symplectic Involution. The Triangular Structure of Ladder Functors.
The Weil Algebra of a Hopf Algebra I: A Noncommutative Framework
Dubois-Violette, Michel; Landi, Giovanni
2014-03-01
We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra in a graded differential algebra Ω. We define the notion of an operation of a Hopf algebra in a graded differential algebra Ω which is referred to as a -operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra of the Hopf algebra is the universal initial object of the category of -operations with connections.
ON THE PRIMARY DECOMPOSITION THEOREM OF MODULAR LIE SUPERALGEBRAS
Institute of Scientific and Technical Information of China (English)
CHEN LIANGYUN; MENG DAOJI
2005-01-01
This gives some identities of associative Lie superalgebras and some properties of modular Lie superalgebras. Furthermore, the primry decomposition theorem of modular Lie superalgebras is shown. It is well known that the primary decomposition theorem of modular Lie algebras has played an important role in the classification of the finite-dimensional simple modular Lie algebras (see [5, 6]). Analogously, the primary decomposition theorem of modular Lie superalgebras may play an important role in the open classification of the finite dimensional simple modular Lie superalgebras.
Topological ∗-algebras with *-enveloping Algebras II
Indian Academy of Sciences (India)
S J Bhatt
2001-02-01
Universal *-algebras *() exist for certain topological ∗-algebras called algebras with a *-enveloping algebra. A Frechet ∗-algebra has a *-enveloping algebra if and only if every operator representation of maps into bounded operators. This is proved by showing that every unbounded operator representation , continuous in the uniform topology, of a topological ∗-algebra , which is an inverse limit of Banach ∗-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-* algebra () of . Given a *-dynamical system (, , ), any topological ∗-algebra containing (, ) as a dense ∗-subalgebra and contained in the crossed product *-algebra *(, , ) satisfies ()=*(, , ). If $G = \\mathbb{R}$, if is an -invariant dense Frechet ∗-subalgebra of such that () = , and if the action on is -tempered, smooth and by continuous ∗-automorphisms: then the smooth Schwartz crossed product $S(\\mathbb{R}, B, )$ satisfies $E(S(\\mathbb{R}, B, )) = C^*(\\mathbb{R}, A, )$. When is a Lie group, the ∞-elements ∞(), the analytic elements () as well as the entire analytic elements () carry natural topologies making them algebras with a *-enveloping algebra. Given a non-unital *-algebra , an inductive system of ideals is constructed satisfying $A = C^*-\\mathrm{ind} \\lim I_$; and the locally convex inductive limit $\\mathrm{ind}\\lim I_$ is an -convex algebra with the *-enveloping algebra and containing the Pedersen ideal of . Given generators with weakly Banach admissible relations , we construct universal topological ∗-algebra (, ) and show that it has a *-enveloping algebra if and only if (, ) is *-admissible.
Cohomology of Heisenberg Lie superalgebras
Bai, Wei; Liu, Wende
2017-02-01
Suppose the ground field to be algebraically closed and of characteristic different from 2 and 3. All Heisenberg Lie superalgebras consist of two super-versions of the Heisenberg Lie algebras, 𝔥2m,n and 𝔟𝔞n with m a non-negative integer and n a positive integer. The space of a "classical" Heisenberg Lie superalgebra 𝔥2m,n is the direct sum of a superspace with a non-degenerate anti-supersymmetric even bilinear form and a one-dimensional space of values of this form constituting the even center. The other super-analog of the Heisenberg Lie algebra, 𝔟𝔞n, is constructed by means of a non-degenerate anti-supersymmetric odd bilinear form with values in the one-dimensional odd center. In this paper, we study the cohomology of 𝔥2m,n and 𝔟𝔞n with coefficients in the trivial module by using the Hochschild-Serre spectral sequences relative to a suitable ideal. In the characteristic zero case, for any Heisenberg Lie superalgebra, we determine completely the Betti numbers and associative superalgebra structures for their cohomology. In the characteristic p > 3 case, we determine the associative superalgebra structure for the divided power cohomology of 𝔟𝔞n and we also make an attempt to determine the divided power cohomology of 𝔥2m,n by computing it in a low-dimensional case.
Some characterizations of Hom-Leibniz algebras
Issa, A Nourou
2010-01-01
Some basic properties of Hom-Leibniz algebras are found. These properties are the Hom-analogue of corresponding well-known properties of Leibniz algebras. Considering the Hom-Akivis algebra associated to a given Hom-Leibniz algebra, it is observed that the Hom-Akivis identity leads to an additional property of Hom-Leibniz algebras, which in turn gives a necessary and sufficient condition for Hom-Lie admissibility of Hom-Leibniz algebras. A necessary and sufficient condition for Hom-power associativity of Hom-Leibniz algebras is also found.
Finite dimensional quadratic Lie superalgebras
Jarvis, Peter; Yates, Luke
2010-01-01
We consider a special class of Z_2-graded, polynomial algebras of degree 2, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. Based on the factorisation of the enveloping algebra, we derive the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate the method for one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.
Norén, Patrik
2013-01-01
Algebraic statistics brings together ideas from algebraic geometry, commutative algebra, and combinatorics to address problems in statistics and its applications. Computer algebra provides powerful tools for the study of algorithms and software. However, these tools are rarely prepared to address statistical challenges and therefore new algebraic results need often be developed. This way of interplay between algebra and statistics fertilizes both disciplines. Algebraic statistics is a relativ...
Whittaker categories and strongly typical Whittaker modules for Lie superalgebras
Bagci, Irfan; Wiesner, Emilie
2012-01-01
Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.
Quantum Heisenberg--Weyl Algebras
Ballesteros, Angel; Herranz, Francisco J.; Parashar, Preeti
1996-01-01
All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$ are classified and explicitly quantized. The complete list of quantum Heisenberg--Weyl algebras so obtained includes new multiparameter deformations, most of them being of the non-coboundary type.
Structure of the Enveloping Algebras
Directory of Open Access Journals (Sweden)
Č. Burdík
2007-01-01
Full Text Available The adjoint representations of several small dimensional Lie algebras on their universal enveloping algebras are explicitly decomposed. It is shown that commutants of raising operators are generated as polynomials in several basic elements. The explicit form of these elements is given and the general method for obtaining these elements is described.
Nakatsuka, Tomoya; Imabayashi, Etsuko; Matsuda, Hiroshi; Sakakibara, Ryuji; Inaoka, Tsutomu; Terada, Hitoshi
2013-05-01
The purpose of this study was to identify brain atrophy specific for dementia with Lewy bodies (DLB) and to evaluate the discriminatory performance of this specific atrophy between DLB and Alzheimer's disease (AD). We retrospectively reviewed 60 DLB and 30 AD patients who had undergone 3D T1-weighted MRI. We randomly divided the DLB patients into two equal groups (A and B). First, we obtained a target volume of interest (VOI) for DLB-specific atrophy using correlation analysis of the percentage rate of significant whole white matter (WM) atrophy calculated using the Voxel-based Specific Regional Analysis System for Alzheimer's Disease (VSRAD) based on statistical parametric mapping 8 (SPM8) plus diffeomorphic anatomic registration through exponentiated Lie algebra, with segmented WM images in group A. We then evaluated the usefulness of this target VOI for discriminating the remaining 30 DLB patients in group B from the 30 AD patients. Z score values in this target VOI obtained from VSRAD were used as the determinant in receiver operating characteristic (ROC) analysis. Specific target VOIs for DLB were determined in the right-side dominant dorsal midbrain, right-side dominant dorsal pons, and bilateral cerebellum. ROC analysis revealed that the target VOI limited to the midbrain exhibited the highest area under the ROC curves of 0.75. DLB patients showed specific atrophy in the midbrain, pons, and cerebellum. Midbrain atrophy demonstrated the highest power for discriminating DLB and AD. This approach may be useful for determining the contributions of DLB and AD pathologies to the dementia syndrome.
EXPANSION DE ALGEBRAS LIE INFINITO-DIMENSIONALES
CAROCA LISBOA, RICARDO ANTONIO
2011-01-01
En Relatividad General el espacio-tiempo es un objeto físico dinámico, que tiene grados de libertad independientes, y es descrito por las ecuaciones de campo de Einstein [1]. Esto significa que en Relatividad General la geometría es descrita de manera dinámica, por lo tanto, la construcción de una teoría de gauge para la gravedad requiere de una acción que no considere un espacio-tiempo background. Una acción para la interacción gravitacional que satisface esta condicion, es dada por un...
M2 to D2 and vice versa by 3-Lie and Lie bialgebra
Energy Technology Data Exchange (ETDEWEB)
Aali-Javanangrouh, M.; Rezaei-Aghdam, A. [Azarbaijan Shahid Madani University, Department of Physics, Faculty of Science, Tabriz (Iran, Islamic Republic of)
2016-11-15
Using the concept of a 3-Lie bialgebra, which has recently been defined in arXiv:1604.04475, we construct a Bagger-Lambert-Gustavson (BLG) model for the M2-brane on a Manin triple of a special 3-Lie bialgebra. Then by using the correspondence and the relation between those 3-Lie bialgebra with Lie bialgebra, we reduce this model to an N = (4,4) WZW model (D2-brane), such that its algebraic structure is a Lie bialgebra with one 2-cocycle. In this manner by using the correspondence of the 3-Lie bialgebra and Lie bialgebra (for this special 3-Lie algebra) one can construct the M2-brane from a D2-brane and vice versa. (orig.)
Induced Modules of Restricted Lie Superalgebras
Institute of Scientific and Technical Information of China (English)
刘文德
2005-01-01
In this paper we first prove the PBW theorem for reduced universal enveloping algebras of restricted Lie superalgebras. Then the notion of an induced module is introduced and the dimension formula of induced modules is established.Finally, using the results above, we obtain a property of induced modules pertaining to automorphisms of Lie superalgebras and isomorphisms of modules.
Gradings and Symmetries on Heisenberg type algebras
A. Calderón; C. Draper; Martín, C.; Sánchez, T.
2014-01-01
We describe the fine (group) gradings on the Heisenberg algebras, on the Heisenberg superalgebras and on the twisted Heisenberg algebras. We compute the Weyl groups of these gradings. Also the results obtained respect to Heisenberg superalgebras are applied to the study of Heisenberg Lie color algebras.
New Matrix Loop Algebra and Its Application
Institute of Scientific and Technical Information of China (English)
DONG Huan-He; XU Yue-Cai
2008-01-01
A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly, as its appfication, the multi-component TC equation hierarchy is obtained, then by use of trace identity the Hamiltonian structure of the above system is presented. Finally, the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.
Central extensions of Lax operator algebras
Schlichenmaier, Martin; Sheinman, Oleg K.
2007-01-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of I.Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this article local cocycles and associated almost-graded central extensions are classified. It is shown that in the case that the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebr...
Lefschetz, Solomon
2005-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
Hom-quadri-代数和 Hom-octo-代数%Hom-Quadri-Algebras and Hom-Octo-Algebras
Institute of Scientific and Technical Information of China (English)
安慧辉; 薛晨; 康健
2014-01-01
Hom-quadri-algebras and Hom-octo-algebras which are respectively motivated from the quadric-algebras and octo-algebras were mainly dicussed. First, some basic definitions of Hom-associative-algebras, Hom-pre-Lie-algebras, Hom-dendriform algebras and Loday algebras were introduced. Then the definition and the structure of Hom-quadri-algebras and Hom-octo-algebras from quadric-algebras and octo-algebra were given. Finally, the relationships among Hom-octo-algebras, Hom-quadri-algebras, Hom-associative-algebras, Hom-pre-Lie-algebras, Hom-dendriform algebras were discussed.%主要研究 Hom-quadri-代数和 Hom-octo-代数，它们分别是由 quadri-代数和 octo-代数通过代数形变得出的。首先给出了 Hom-associative-代数、Hom-pre-Lie-代数，Hom-dendriform 代数和 Loday 代数的一些基本定义和性质，然后将 quadri-代数、octo-代数的定义推广到了 Hom-quadri-代数、Hom-octo-代数，最后研究并论证了 Hom-quadri-代数和 Hom-octo-代数与 Hom-associative-代数、Hom-pre-Lie-代数、Hom-dendriform 代数之间的关系。
A Hopf algebra deformation approach to renormalization
Ionescu, L M; Ionescu, Lucian M.; Marsalli, Michael
2003-01-01
We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices. It is suggested that the QFTs obtained via deformation quantization and renormalization correspond to each other in the sense of Kontsevich/Cattaneo-Felder.
Beilinson, Alexander
2004-01-01
Chiral algebras form the primary algebraic structure of modern conformal field theory. Each chiral algebra lives on an algebraic curve, and in the special case where this curve is the affine line, chiral algebras invariant under translations are the same as well-known and widely used vertex algebras. The exposition of this book covers the following topics: the "classical" counterpart of the theory, which is an algebraic theory of non-linear differential equations and their symmetries; the local aspects of the theory of chiral algebras, including the study of some basic examples, such as the ch
The Algebra of Conformal Blocks
Manon, Christopher A
2009-01-01
We study and generalize the connection between the phylogenetic Hilbert functions of Buczynska and Wisniewski \\cite{BW} and the Verlinde formula, as discovered by Sturmfels and Xu in \\cite{StXu}. In order to accomplish this we introduce deformations of algebras of non-abelian theta functions for a general simple complex Lie algebra $\\mathfrak{g}$ structured on the moduli stack of stable punctured curves. We also study the relationship between these algebras and branching algebras, coming from the representation theory of the associated reductive group $G.$
String Quantization and the Shuffle Hopf Algebra
Bahns, Dorothea
2011-01-01
The Poisson algebra $\\mathfrak h$ of invariants of the Nambu-Goto string, which was first introduced by K. Pohlmeyer in 1982, is described using the Shuffle Hopf algebra. In particular, an underlying auxiliary Lie algebra is reformulated in terms of the image of the first Eulerian idempotent of the Shuffle Hopf algebra. This facilitates the comparison of different approaches to the quantization of $\\mathfrak h$.
Implications of the Hopf algebra properties of noncommutative differential calculi
1996-01-01
We define a noncommutative algebra of four basic objects within a differential calculus on quantum groups: functions, 1-forms, Lie derivatives and inner derivations, as the cross-product algebra associated with Woronowicz's (differential) algebra of functions and forms. This definition properly takes into account the Hopf algebra structure of the Woronowicz calculus. It also provides a direct proof of the Cartan identity.
Implications of the Hopf algebra properties of noncommutative differential calculi
Vladimirov, A.A.
1996-01-01
We define a noncommutative algebra of four basic objects within a differential calculus on quantum groups: functions, 1-forms, Lie derivatives and inner derivations, as the cross-product algebra associated with Woronowicz's (differential) algebra of functions and forms. This definition properly takes into account the Hopf algebra structure of the Woronowicz calculus. It also provides a direct proof of the Cartan identity.
Affine and degenerate affine BMW algebras: Actions on tensor space
Daugherty, Zajj; Virk, Rahbar
2012-01-01
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements--the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.
Classification of real low-dimensional Jacobi (generalized)-Lie bialgebras
Rezaei-Aghdam, A.; Sephid, M.
2017-09-01
We describe the definition of Jacobi (generalized)-Lie bialgebras ((g,ϕ0), (g∗,X 0)) in terms of structure constants of the Lie algebras g and g∗ and components of their 1-cocycles X0 ∈g and ϕ0 ∈g∗ in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi-Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi-Lie bialgebras.
On indecomposable modules over the Virasoro algebra
Institute of Scientific and Technical Information of China (English)
sU; Yucai(
2001-01-01
［1］Chari, V. , Pressley, A., Unitary representations of the Virasoro algebra and a conjecture of Kac, Compositio Math, 1988,67: 315-342.［2］Feign, B. L. , Fuchs, D. B., Verma modules over the Virasoro algebra, Lecture Notes in Math, 1984, 1060: 230-245.［3］Kac, V. G., Some problems on infinite-dimensional Lie algebras and their representations, Lie algebras and related topics,Lecture Notes in Math., 1982, 933: 117-126.［4］Kac, V. G., Infinite Dimensional Lie Algebras, 2nd ed., Boston, Cambridge: Birkhauser, 1985.［5］Kaplansky, I., Santharoubane, L. J., Harish-Chandra modules over the Virasoro algebra, Infinite-dimensional groups with application, Math. Sci. Res. Inst. Pub., 1985, 4: 217-231.［6］Langlands, R., On unitary representations of the Virasoro algebra, Infinite-Dimensional Lie Algebras and Their Application,Singapore: World Scientific, 1986, 141-159.［7］Martin. C. , Piard, A., Indecomposable modules over the Virasoro Lie algebra and a conjecture of V Kac, Comm. Math.Phys., 1991, 137: 109-132.［8］Mathieu, O. , Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent Math., 1992, 107: 225-234.［9］Su, Y., A classification of indecomposable sl2(2)-modules and a conjecture of Kac on irreducible modules over the Virasoro algebra, J. Alg., 1993, 161: 33-46.［10］Su, Y. , Classification of Harish-Chandra modules over the super-Virasoro algebras, Comm. Alg., 1995, 23: 3653-3675.［11］Su, Y. , Simple modules over the high rank Virasoro algebras, Comm. Alg., 2001, in press.
On n-ary Hom-Nambu and Hom-Maltsev algebras
Yau, Donald
2010-01-01
Hom-alternative and Hom-Jordan algebras are shown to give rise to Hom-Nambu algebras of arities 2^{k+1} + 1. The class of n-ary Hom-Maltsev algebras is studied. Multiplicative n-ary Hom-Nambu-Lie algebras are shown to be n-ary Hom-Maltsev algebras. Examples of ternary Hom-Maltsev algebras that are not ternary Hom-Nambu-Lie algebras are given. Ternary Hom-Maltsev algebras are shown to arise from composition algebras.
Completely Positive Definite Maps on σ-C*-algebras
Institute of Scientific and Technical Information of China (English)
许天周; 段培超; 郑庆琳
2003-01-01
@@ Recently, there has been increased interest [1-8] in topological *-algebras that are inverselimits of C*-algebras, called Pro-C*-algebras. These algebras were introduced in [5] as a gene-ralization of C*-algebras were called locally C*-algebras. The same objects have been studiedvarious term, in [1-8], it is shown in [6-7] that they arise naturally in certain aspects of C*-algebraslike the tangent algebras of C*-algebras, multipliers of Pedersen's ideal, non-commutative ana-logues of classical Lie groups and K-theory.
On the projective algebra of Einstein Matsumoto metrics
Rafie-Rad, M
2011-01-01
The projective algebra p(M;F) (i.e the collection of all projective vector fields)of a Finsler space (M;F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket. The projective algebra of Einstein metrics has been perpetually studied from physical and geometrical approaches. Here, the projective algebra of Einstein Matsumoto space of dimension n \\geq 3 is characterized. Moreover, Einstein Matsumoto metrics with maximum projective symmetry are studied and characterized.