Computations in finite-dimensional Lie algebras
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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.
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.
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
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.
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.
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.
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.
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.
Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras
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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.
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
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.
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.
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.
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 .
Weak Lie symmetry and extended Lie algebra
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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).
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.
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.
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.
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...
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
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.
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).
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.
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.
Directory of Open Access Journals (Sweden)
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.
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.
Semiclassical states on Lie algebras
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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.
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.
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.
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).
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.
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
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.
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...
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.
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.
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.
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.
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.
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.
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...
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.
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.
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.
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.
Loop Virasoro Lie conformal algebra
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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.
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 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.
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.
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.
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.
The Lie algebra of the N=2-string
Energy Technology Data Exchange (ETDEWEB)
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.)
Institute of Scientific and Technical Information of China (English)
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.
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 ...
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.
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.
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
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.
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.
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 $[S^-,S^-]$.
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.
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 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.
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.
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.
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.
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.
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.
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.
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...
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.
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.
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 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}.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Finite-dimensional (*)-serial algebras
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let A be a finite-dimensional associative algebra with identity over a field k. In this paper we introduce the concept of (*)-serial algebras which is a generalization of serial algebras. We investigate the properties of (*)-serial algebras, and we obtain suficient and necessary conditions for an associative algebra to be (*)-serial.
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.
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.
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...
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.
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.
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).
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+).
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.
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.
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.
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.
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.
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.
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.
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.
Sectional and Ricci Curvature for Three-Dimensional Lie Groups
Directory of Open Access Journals (Sweden)
Gerard Thompson
2016-01-01
Full Text Available Formulas for the Riemann and Ricci curvature tensors of an invariant metric on a Lie group are determined. The results are applied to a systematic study of the curvature properties of invariant metrics on three-dimensional Lie groups. In each case the metric is reduced by using the automorphism group of the associated Lie algebra. In particular, the maximum and minimum values of the sectional curvature function are determined.
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.
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 $g$.
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.
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.
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.
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.
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.
Finite-dimensional representations of twisted hyper loop algebras
Bianchi, Angelo
2012-01-01
We investigate the category of finite-dimensional representations of twisted hyper loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper loop algebras are isomorphic to appropriate simple and Weyl modules for the non-twisted hyper loop algebras, respectively, via restriction of the action.
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.
On q-deformed infinite-dimensional n-algebra
Directory of Open Access Journals (Sweden)
Lu Ding
2016-03-01
Full Text Available The q-deformation of the infinite-dimensional n-algebras is investigated. Based on the structure of the q-deformed Virasoro–Witt algebra, we derive a nontrivial q-deformed Virasoro–Witt n-algebra which is nothing but a sh-n-Lie algebra. Furthermore in terms of the pseud-differential operators, we construct the (cosine n-algebra and the q-deformed SDiff(T2 n-algebra. We find that they are the sh-n-Lie algebras for the n even case. In terms of the magnetic translation operators, an explicit physical realization of the (cosine n-algebra is given.
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.
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...
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.
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.
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.
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.
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.
Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics
Energy Technology Data Exchange (ETDEWEB)
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
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)
2007-05-18
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.
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 ...
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.
A Lie based 4-dimensional higher Chern-Simons theory
Zucchini, Roberto
2015-01-01
We present and study a model of 4-dimensional higher Chern-Simons theory, special Chern-Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2-algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2-connection coupled to a background closed 3-form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2-group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3-form. Finally, SCS theory is related to a 3-dimensional special gauge theory whose 2-connection space has a natural symplectic structure with respect to which the 1-gauge transformation action is Hamiltonian, the 2-curvature map acting as moment map.
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.
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.
Infinite-dimensional Hamiltonian Lie superalgebras
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
The natural filtration of the infinite-dimensional Hamiltonian Lie superalgebra over a field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements.We are thereby able to obtain an intrinsic characterization of the Hamiltonian Lie superalgebra and establish a property of the automorphisms of the Lie superalgebra.
Alternative description of three dimensional complex diassociative algebras with some constraints
Rikhsiboev, Ikrom M.; Venkatesan, Yuvendra Rao
2014-07-01
Considering significant of classification problems in modern algebra, especially in the algebras which related to Lie algebras, current research pursue investigation on structure theory of low dimensional diassociative algebras. Note that the classification of complex diassociative algebras in low dimensions have been presented in our recent studies, however this paper deals to provide description of such algebras with some constrains in dimension three, applying notion of annihilator.
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.
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.
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.
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...
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.
Finite dimensional quotients of commutative operator algebras
Meyer, Ralf
1997-01-01
The matrix normed structure of the unitization of a (non-selfadjoint) operator algebra is determined by that of the original operator algebra. This yields a classification up to completely isometric isomorphism of two-dimensional unital operator algebras. This allows to define invariant distances on the spectrum of commutative operator algebras analogous to the Caratheodory distance for complex manifolds. Moreover, unitizations of two-dimensional operator algebras with zero multiplication pro...
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.
A Note on Four-Dimensional Symmetry Algebras and Fourth-Order Ordinary Differential Equations
Directory of Open Access Journals (Sweden)
A. Fatima
2013-01-01
Full Text Available We provide a supplementation of the results on the canonical forms for scalar fourth-order ordinary differential equations (ODEs which admit four-dimensional Lie algebras obtained recently. Together with these new canonical forms, a complete list of scalar fourth-order ODEs that admit four-dimensional Lie algebras is available.
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.
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.
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.
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
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
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)
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.
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).
Finite-dimensional division algebras over fields
Jacobson, Nathan
2009-01-01
Finite-Dimensional Division Algebras over fields determine, by the Wedderburn Theorem, the semi-simple finite-dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brauer-Severi varieties. The book concentrates on those algebras that have an involution. Algebras with involution appear in many contexts; they arose first in the study of the so-called 'multiplication algebras of Riemann matrices'. The largest part of the book is the fifth chapter, dealing with involutorial simple algebras of finite dimension over a field. Of parti
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...
Finite dimensional quotients of commutative operator algebras
Meyer, R
1997-01-01
The matrix normed structure of the unitization of a (non-selfadjoint) operator algebra is determined by that of the original operator algebra. This yields a classification up to completely isometric isomorphism of two-dimensional unital operator algebras. This allows to define invariant distances on the spectrum of commutative operator algebras analogous to the Caratheodory distance for complex manifolds. Moreover, unitizations of two-dimensional operator algebras with zero multiplication provide a rich class of counterexamples. Especially, several badly behaved quotients of function algebras are exhibited. Recently, Arveson has developed a model theory for d-contractions. Quotients of the operator algebra of the d-shift are much more well-behaved than quotients of function algebras. Completely isometric representations of these quotients are obtained explicitly. This provides a generalization of Nevanlinna-Pick theory. An important property of quotients of the d-shift algebra is that their quotients of finit...
Classification of 5-Dimensional MD-Algebras Having Non-Commutative Derived Ideals
Vu, Le Anh; Nghia, Tran Thi Hieu
2011-01-01
The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal dimension. The main result of the paper is the classification up to an isomorphism of all MD5-algebras with the non-commutative derived ideal. With this result, we have the complete classification of 5-dimensional solvable Lie algebras.
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.
Imai, Kenji
2014-02-01
In this paper, a new n-dimensional homogeneous Lotka-Volterra (HLV) equation, which possesses a Lie symmetry, is derived by the extension from a three-dimensional HLV equation. Its integrability is shown from the viewpoint of Lie symmetries. Furthermore, we derive dynamical systems of higher order, which possess the Lie symmetry, using the algebraic structure of this HLV equation.
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.
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.
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}.
Three-dimensional polarization algebra.
R Sheppard, Colin J; Castello, Marco; Diaspro, Alberto
2016-10-01
If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.
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.
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.
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.
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。
Equivalency of two-dimensional algebras
Energy Technology Data Exchange (ETDEWEB)
Santos, Gildemar Carneiro dos; Pomponet Filho, Balbino Jose S. [Universidade Federal da Bahia (UFBA), BA (Brazil). Inst. de Fisica
2011-07-01
Full text: Let us consider a vector z = xi + yj over the field of real numbers, whose basis (i,j) satisfy a given algebra. Any property of this algebra will be reflected in any function of z, so we can state that the knowledge of the properties of an algebra leads to more general conclusions than the knowledge of the properties of a function. However structural properties of an algebra do not change when this algebra suffers a linear transformation, though the structural constants defining this algebra do change. We say that two algebras are equivalent to each other whenever they are related by a linear transformation. In this case, we have found that some relations between the structural constants are sufficient to recognize whether or not an algebra is equivalent to another. In spite that the basis transform linearly, the structural constants change like a third order tensor, but some combinations of these tensors result in a linear transformation, allowing to write the entries of the transformation matrix as function of the structural constants. Eventually, a systematic way to find the transformation matrix between these equivalent algebras is obtained. In this sense, we have performed the thorough classification of associative commutative two-dimensional algebras, and find that even non-division algebra may be helpful in solving non-linear dynamic systems. The Mandelbrot set was used to have a pictorial view of each algebra, since equivalent algebras result in the same pattern. Presently we have succeeded in classifying some non-associative two-dimensional algebras, a task more difficult than for associative one. (author)
SOME RESULTS ON INFINITE DIMENSIONAL NOVIKOV ALGEBRAS
Institute of Scientific and Technical Information of China (English)
赵玉凤; 孟道骥
2003-01-01
This paper gives some sufficient conditions for determining the simplicity of infinite di-mensional Novikov algebras of characteristic 0, and also constructs a class of simple Novikovalgebras by extending the base field. At last, the deformation theory of Novikov algebras isintroduced.
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.
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)
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.
A Higher Dimensional Loop Algebra and Integrable Couplings System of Evolution Equations Hierarchy
Institute of Scientific and Technical Information of China (English)
夏铁成; 于发军; 陈登远
2005-01-01
An extension of the Lie algebra An-1 has been proposed [ Phys. Lett. A, 2003, 310 : 19-24 ]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra G～. Based on the loop algebra G～, the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.
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...
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 ...
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.
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 \
Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras
Directory of Open Access Journals (Sweden)
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].
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.
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.
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 ...
Falomir, H.; Pisani, P. A. G.; Vega, F.; Cárcamo, D.; Méndez, F.; Loewe, M.
2016-02-01
We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras {sl}(2,{{R}}) or su(2) according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, such as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.
Hypercontractivity in finite-dimensional matrix algebras
Energy Technology Data Exchange (ETDEWEB)
Junge, Marius, E-mail: junge@math.uiuc.edu [Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61891 (United States); Palazuelos, Carlos, E-mail: carlospalazuelos@ucm.es [Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Plaza de Ciencias s/n, 28040 Madrid (Spain); Parcet, Javier, E-mail: javier.parcet@icmat.es; Perrin, Mathilde, E-mail: mathilde.perrin@icmat.es [Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 13-15, 28049 Madrid (Spain)
2015-02-15
We obtain hypercontractivity estimates for a large class of semigroups defined on finite-dimensional matrix algebras M{sub n}. These semigroups arise from Poisson-like length functions ψ on ℤ{sub n} × ℤ{sub n} and provide new hypercontractive families of quantum channels when ψ is conditionally negative. We also study the optimality of our estimates.
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.
A (1 + 2-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II
Directory of Open Access Journals (Sweden)
Roman Cherniha
2017-01-01
Full Text Available A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1-dimensional. Exact solutions of some (1 + 1-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found.
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.
Classification of Low-Dimensional Hom-Leibniz Algebras%低维Hom-Leibniz代数分类
Institute of Scientific and Technical Information of China (English)
徐丽媛; 王春月; 张若兰; 张庆成
2013-01-01
运用待定系数法确定了复数域上的二维和三维Leibniz代数的自同态,进而对相关非李代数的Hom-Leibniz代数进行了分类.%The authors determined the two dimensional and three dimensional endomorphism of Leibniz algebras on complex field using undetermined coefficients method, and then clssified the Hom-Leibniz algebras of their associated non-Lie algebra.
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\
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 ...
Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems
Fazlul Hoque, Md; Marquette, Ian; Zhang, Yao-Zhong
2016-03-01
By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the N-dimensional superintegrable Kepler-Coulomb model with non-central terms and the double singular oscillators of type (n,N-n). We show how the integrals of motion generate higher rank cubic algebra C(3)\\oplus {L}1\\oplus {L}2 with structure constants involving Casimir operators of the Lie algebras L 1 and L 2. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.
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).
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.
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.
Aspects of the Supersymmetry Algebra in Four Dimensional Euclidean Space
McKeon, D G C
1998-01-01
The simplest supersymmetry (SUSY) algebra in four dimensional Euclidean space ($4dE$) has been shown to closely resemble the $N = 2$ SUSY algebra in four dimensional Minkowski space ($4dM$). The structure of the former algebra is examined in greater detail in this paper. We first present its Clifford algebra structure. This algebra shows that the momentum Casimir invariant of physical states has an upper bound which is fixed by the central charges. Secondly, we use reduction of the $N = 1$ SUSY algebra in six dimensional Minkowski space ($6dM$) to $4dE$; this reproduces our SUSY algebra in $4dE$. Moreover, this same reduction of supersymmetric Yang-Mills theory (SSYM) in $6dM$ reproduces Zumino's SSYM in $4dE$. We demonstrate how this dimensional reduction can be used to introduce additional generators into the SUSY algebra in $4dE$.
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.
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...
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.
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.
Cohomology spaces of low dimensional complex associative algebras
Mohammed, Nadia F.; Rakhimov, Isamiddin S.; Hussain, Sharifah Kartini Said
2017-04-01
In this paper, we calculate cohomology groups of low-dimensional complex associative algebras. The calculations are based on a classification result and description of derivations of low-dimensional associative algebras obtained earlier. For the first cohomology group, we give basic cocycles up to inner derivations. We also provide basic coboundaries for the second cohomology groups for low-dimensional associative algebras (including both unital and non unital).
Falomir, H; Vega, F; Cárcamo, D; Méndez, F; Loewe, M
2015-01-01
We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras $sl(2,\\mathbb{R})$ or $su(2)$ according to the relation between the noncommutativity parameters. From this perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.
The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator
Energy Technology Data Exchange (ETDEWEB)
Borzov, V. V., E-mail: borzov.vadim@yandex.ru [Department of Mathematics, St. Petersburg State University of Telecommunications, 191186, Moika 61, St. Petersburg (Russian Federation); Damaskinsky, E. V., E-mail: evd@pdmi.ras.ru [Department of Natural Sciences, Institute of Defense Technical Engineering (VITI), 191123, Zacharievskaya 22, St. Petersburg (Russian Federation)
2014-10-15
In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys. 175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
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.
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.
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
Directory of Open Access Journals (Sweden)
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.
Directory of Open Access Journals (Sweden)
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.
Regularity properties of infinite-dimensional Lie groups, and semiregularity
Glockner, Helge
2012-01-01
Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is C^k-semiregular if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map taking c to evol(c):=Evol(c)(1) is smooth, then G is called C^k-regular. For G a C^k-semiregular Lie group and m an order of differentiability, we show that evol is C^m if and only if Evol is C^m. If evol is continuous at 0, then evol is continuous. If G is...
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
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.
Abedi-Fardad, J.; Rezaei-Aghdam, A.; Haghighatdoost, Gh.
2017-01-01
We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson-Lie groups. We obtain some new integrable models where a Poisson-Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.
Aspects of infinite dimensional ℓ-super Galilean conformal algebra
Aizawa, N.; Segar, J.
2016-12-01
In this work, we construct an infinite dimensional ℓ-super Galilean conformal algebra, which is a generalization of the ℓ = 1 algebra found in the literature. We give a classification of central extensions, the vector field representation, the coadjoint representation, and the operator product expansion of the infinite dimensional ℓ-super Galilean conformal algebra, keeping possible applications in physics and mathematics in mind.
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.
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。
A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras
Koç, Ayten; Güloğlu, Ismail; Kanuni, Müge; Koc, Ayten; Esin, Songul; Guloglu, Ismail; Kanuni, Muge
2012-01-01
Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant {\\kappa}(A) for A and count the number of isomorphism classes of Leavitt path algebras with {\\kappa}(A)=n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.
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.
Classical Lie Point Symmetry Analysis of a Steady Nonlinear One-Dimensional Fin Problem
Directory of Open Access Journals (Sweden)
R. J. Moitsheki
2012-01-01
Full Text Available We consider the one-dimensional steady fin problem with the Dirichlet boundary condition at one end and the Neumann boundary condition at the other. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended. Some invariant solutions are constructed. The effects of thermogeometric fin parameter and the exponent on temperature are studied. Also, the fin efficiency is analyzed.
Finite dimensional semigroup quadratic algebras with minimal number of relations
Iyudu, Natalia
2011-01-01
A quadratic semigroup algebra is an algebra over a field given by the generators $x_1,...,x_n$ and a finite set of quadratic relations each of which either has the shape $x_jx_k=0$ or the shape $x_jx_k=x_lx_m$. We prove that a quadratic semigroup algebra given by $n$ generators and $d\\leq \\frac{n^2+n}{4}$ relations is always infinite dimensional. This strengthens the Golod--Shafarevich estimate for the above class of algebras. Our main result however is that for every $n$, there is a finite dimensional quadratic semigroup algebra with $n$ generators and $\\delta_n$ generators, where $\\delta_n$ is the first integer greater than $\\frac{n^2+n}{4}$. This shows that the above Golod-Shafarevich type estimate for semigroup algebras is sharp.
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.
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.
Invariants of 3-Manifolds derived from finite dimensional hopf algebras
Kauffman, L H; Louis H Kauffman; David E Radford
1994-01-01
Abstract: This paper studies invariants of 3-manifolds derived from certain fin ite dimensional Hopf algebras. The invariants are based on right integrals for these algebras. It is shown that the resulting class of invariants is distinct from the class of Witten-Reshetikhin-Turaev invariants.
An algebraic study of unitary one dimensional quantum cellular automata
Arrighi, P
2005-01-01
We provide algebraic characterizations of unitary one dimensional quantum cellular automata. We do so both by algebraizing existing decision procedures, and by adding constraints into the model which do not change the quantum cellular automata's computational power. The configurations we consider have finite but unbounded size.
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...
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.
Algebraic curves and one-dimensional fields
Bogomolov, Fedor
2002-01-01
Algebraic curves have many special properties that make their study particularly rewarding. As a result, curves provide a natural introduction to algebraic geometry. In this book, the authors also bring out aspects of curves that are unique to them and emphasize connections with algebra. This text covers the essential topics in the geometry of algebraic curves, such as line and vector bundles, the Riemann-Roch Theorem, divisors, coherent sheaves, and zeroth and first cohomology groups. The authors make a point of using concrete examples and explicit methods to ensure that the style is clear an
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...
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...
Operator spaces and residually finite-dimensional $C^{*}$-algebras
Pestov, V G
1993-01-01
For every operator space $X$ the $C^\\ast$-algebra containing it in a universal way is residually finite-dimensional (that is, has a separating family of finite-dimensional representations). In particular, the free $C^\\ast$-algebra on any normed space so is. This is an extension of an earlier result by Goodearl and Menal, and our short proof is based on a criterion due to Exel and Loring.
Classical r-matrices of real low-dimensional Jacobi-Lie bialgebras and their Jacobi-Lie groups
Rezaei-Aghdam, A.; Sephid, M.
2014-01-01
In this research we obtain the classical r-matrices of real two and three dimensional Jacobi-Lie bialgebras. In this way, we classify all non-isomorphic real two and three dimensional coboundary Jacobi-Lie bialgebras and their types (triangular and quasitriangular). Also, we obtain the generalized Sklyanin bracket formula by use of which, we calculate the Jacobi structures on the related Jacobi-Lie groups. Finally, we present a new method for constructing classical integrable systems using co...
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 $\\le 5$.
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.
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.
Diffeomorphisms from higher dimensional W-algebras
Martínez-Moras, F; Ramos, E; Moras, Fernando Martinez; Mas, Javier; Ramos, Eduardo
1993-01-01
Classical W-algebras in higher dimensions have been recently constructed. In this letter we show that there is a finitely generated subalgebra which is isomorphic to the algebra of local diffeomorphisms in D dimensions. Moreover, there is a tower of infinitely many fields transforming under this subalgebra as symmetric tensorial one-densities. We also unravel a structure isomorphic to the Schouten symmetric bracket, providing a natural generalization of w_\\infty in higher dimensions.
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.
Algebraic surface grid generation in three-dimensional space
Warsi, Saif
1992-01-01
An interactive program for algebraic generation of structured surface grids in three dimensional space was developed on the IRIS4D series workstations. Interactive tools are available to ease construction of edge curves and surfaces in 3-D space. Addition, removal, or redistribution of points at arbitrary locations on a general 3-D surface or curve is possible. Also, redistribution of surface grid points may be accomplished through use of conventional surface splines or a method called 'surface constrained transfinite interpolation'. This method allows the user to redistribute the grid points on the edges of a surface patch; the effect of the redistribution is then propagated to the remainder of the surface through a transfinite interpolation procedure where the grid points will be constrained to lie on the surface. The program was written to be highly functional and easy to use. A host of utilities are available to ease the grid generation process. Generality of the program allows the creation of single and multizonal surface grids according to the user requirements. The program communicates with the user through popup menus, windows, and the mouse.
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.
The Socle and Finite Dimensionality of some Banach Algebras
Indian Academy of Sciences (India)
Ali Ghaffari; Ali Reza Medghalchi
2005-08-01
The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact group , is compact if there exists a measure in $\\mathrm{Soc} (L^1(G))$ such that () ≠ 0. We also prove that is finite if $\\mathrm{Soc}(M(G))$ is closed and every nonzero left ideal in () contains a minimal left ideal.
Two dimensional basic linear algebra communication subprograms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Whaley, R.C. [Univ. of Tennessee, Knoxville, TN (United States); Geijn, R.A. van de [Univ. of Texas, Austin, TX (United States)
1993-12-31
This paper describes a package of linear algebra communication routines for manipulating and communicating data structures that are distributed among the memories of a distributed memory MIMD computer. The motivation for the BLACS is to increase portability, efficiency and modularity at a high level. The audience of the BLACS are mathematical software experts and people with large scale scientific computation to perform.
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.
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.
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.
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.
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.
Family of N-dimensional superintegrable systems and quadratic algebra structures
Fazlul Hoque, Md; Marquette, Ian; Zhang, Yao-Zhong
2016-01-01
Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in pure and applied mathematics. We overview two new families of superintegrable Kepler-Coulomb systems with non-central terms and superintegrable Hamiltonians with double singular oscillators of type (n, N — n) in N-dimensional Euclidean space. We present their quadratic and polynomial algebras involving Casimir operators of so(N + 1) Lie algebras that exhibit very interesting decompositions Q(3) ⊕ so(N — 1), Q(3) ⊕ so(n) ⊕ so(N — n) and the cubic Casimir operators. The realization of these algebras in terms of deformed oscillator enables the determination of a finite dimensional unitary representation. We present algebraic derivations of the degenerate energy spectra of these systems and relate them with the physical spectra obtained from the separation of variables.
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.
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.
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.
Deformed oscillator algebras for two dimensional quantum superintegrable systems
Bonatsos, Dennis; Kokkotas, K D; Bonatsos, Dennis
1994-01-01
Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions.
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.
Private algebras in quantum information and infinite-dimensional complementarity
Energy Technology Data Exchange (ETDEWEB)
Crann, Jason, E-mail: jason-crann@carleton.ca [School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6 (Canada); Laboratoire de Mathématiques Paul Painlevé–UMR CNRS 8524, UFR de Mathématiques, Université Lille 1–Sciences et Technologies, 59655 Villeneuve d’Ascq Cédex (France); Kribs, David W., E-mail: dkribs@uoguelph.ca [Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1 (Canada); Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada); Levene, Rupert H., E-mail: rupert.levene@ucd.ie [School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4 (Ireland); Todorov, Ivan G., E-mail: i.todorov@qub.ac.uk [Pure Mathematics Research Centre, Queen’s University Belfast, Belfast BT7 1NN (United Kingdom)
2016-01-15
We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.
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.
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.
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.
Wigner distributions for finite dimensional quantum systems: An algebraic approach
Indian Academy of Sciences (India)
S Chaturvedi; E Ercolessi; G Marmo; G Morandi; N Mukunbda; R Simon
2005-12-01
We discuss questions pertaining to the definition of `momentum', `momentum space', `phase space' and `Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of `momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.
Institute of Scientific and Technical Information of China (English)
夏铁成; 于发军; 陈登远
2005-01-01
An extension of the Lie algebra An-1 has been proposed [Phys. Lett. A, 2003, 310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra G. Based on the loop algebra G, the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.
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.
An Infinite Dimensional Symmetry Algebra in String Theory
Evans, Mark; Nanopoulos, Dimitri V.; Evans, Mark; Giannakis, Ioannis
1994-01-01
Symmetry transformations of the space-time fields of string theory are generated by certain similarity transformations of the stress-tensor of the associated conformal field theories. This observation is complicated by the fact that, as we explain, many of the operators we habitually use in string theory (such as vertices and currents) have ill-defined commutators. However, we identify an infinite-dimensional subalgebra whose commutators are not singular, and explicitly calculate its structure constants. This constitutes a subalgebra of the gauge symmetry of string theory, although it may act on auxiliary as well as propagating fields. We term this object a {\\it weighted tensor algebra}, and, while it appears to be a distant cousin of the $W$-algebras, it has not, to our knowledge, appeared in the literature before.
Geometrization of $N$-Extended $1$-Dimensional Supersymmetry Algebras II
Doran, Charles; Kostiuk, Jordan; Méndez-Diez, Stefan
2016-01-01
The problem of classifying off-shell representations of the $N$ -extended one-dimensional super Poincar\\'e algebra is closely related to the study of a class of decorated $N$-regular, $N$-edge colored bipartite graphs known as Adinkras. In previous work we canonically embedded these graphs into explicitly uniformized Riemann surfaces via the "dessins d'enfant" construction of Grothendieck. The Adinkra graphs carry two additional structures: a selection of dashed edges and an assignment of integral helghts to the vertices. In this paper, we complete the passage from algebra, through discrete structures, to geometry. We show that the dashings correspond to special spin structures on the Riemann surface, defining thereby super Riemann surfaces. Height assignments determine discrete Morse functions, from which we produce a set of Morse divisors which capture the topological properties of the height assignments
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.
3-Lie bialgebras (Lb,Cd and (Lb,Ce
Directory of Open Access Journals (Sweden)
Bai Ruipu
2016-05-01
Full Text Available Four dimensional $3$-Lie coalgebras with two-dimensional derived algebras, and four-dimensional $3$-Lie bialgebras of type $(L_b, C_c$ are classified. It is proved that there exist three classes of four dimensional $3$-Lie coalgebras with two-dimensional derived algebra which are $(L, C_{c_i}$, $i=1, 2, 3$ (Lemma 3.1, and ten classes of four dimensional $3$-Lie bialgebras of type $(L_b, C_c$ (Theorem 3.2.
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.
Applications of Lie Symmetries to Higher Dimensional Gravitating Fluid
Msomi, A M; Maharaj, S D
2013-01-01
We consider a radiating shear-free spherically symmetric metric in higher dimensions. Several new solutions to the Einstein's equations are found systematically using the method of Lie analysis of differential equations. Using the five Lie point symmetries of the fundamental field equation, we obtain either an implicit solution or we can reduce the governing equations to a Riccati equation. We show that known solutions of the Einstein equations can produce infinite families of new solutions. Earlier results in four dimensions are shown to be special cases of our generalised results.
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.
Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras
Doran, Charles; Landweber, Greg; Mendez-Diez, Stefan
2013-01-01
The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincar\\'{e} algebra is closely related to the study of a class of decorated graphs known as Adinkras. We show that these combinatorial objects possess a form of emergent supergeometry: Adinkras are equivalent to very special super Riemann surfaces with divisors. The method of proof critically involves Grothendieck's theory of "dessins d'enfants'', work of Cimasoni-Reshetikhin expressing spin structures on Riemann surfaces via dimer models, and an observation of Donagi-Witten on parabolic structure from ramified coverings of super Riemann surfaces.
Infinite Dimensional Free Algebra and the Forms of the Master Field
Halpern, M B
1999-01-01
We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and the dual basis is closely related to the planar connected parts. This leads to a number of free-algebraic forms of the master field including an algebraic derivation of the Gopakumar-Gross form. For action theories, these forms of the master field immediately give a number of new free-algebraic packagings of the planar Schwinger-Dyson equations.
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.
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.
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...
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)
Quantum algebras for maximal motion groups of n-dimensional flat spaces
Ballesteros, A; Del Olmo, M A; Santander, M
1994-01-01
An embedding method to get q-deformations for the non-semisimple algebras generating the motion groups of N-dimensional flat spaces is presented. This method gives a global and simultaneous scheme of q-deformation for all iso(p,q) algebras and for those ones obtained from them by some Inönü-Wigner contractions, such as the N--dimensional Euclidean, Poincaré and Galilei algebras.
Algebraic geometry methods associated to the one-dimensional Hubbard model
Energy Technology Data Exchange (ETDEWEB)
Martins, M.J., E-mail: martins@df.ufscar.br
2016-06-15
In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective R-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct J-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the R-matrix can not be written using only difference of spectral parameters of the Lax operator.
Krichever-Novikov type algebras theory and applications
Schlichenmaier, Martin
2014-01-01
Krichever and Novikov introduced certain classes of infinite dimensionalLie algebrasto extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them toa more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are
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.
Real zeros of the zero-dimensional parametric piecewise algebraic variety
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi- algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and suffcient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and suffcient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-01
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
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.
Batalin-Vilkovisky algebras and two-dimensional topological field theories
Getzler, E
1994-01-01
Batalin-Vilkovisky algebras are a new type of algebraic structure on graded vector spaces, which first arose in the work of Batalin and Vilkovisky on gauge fixing in quantum field theory. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological field theory in two dimensions. Lian and Zuckerman have constructed this Batalin-Vilkovisky structure, in the setting of topological chiral field theories, and shown that the structure is non-trivial in two-dimensional string theory. Our approach is to use algebraic topology, whereas their proofs have a more algebraic character.
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.
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.
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.
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.
Geometry of the gauge algebra in noncommutative Yang-Mills theory
Lizzi, Fedele; Zampini, Alessandro; Szabo, Richard J.
2001-08-01
A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in non-commutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra u(∞), and of the C*-algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated.
Geometry of the Gauge Algebra in Noncommutative Yang-Mills Theory
Lizzi, F; Zampini, A
2001-01-01
A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra, and of the algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated.
Vibrational spectroscopy of SnBr4 and CCl4 using Lie algebraic approach
Indian Academy of Sciences (India)
Joydeep Choudhury; Srinivasa Rao Karmuri; Nirmal Kumar Sarkar; Ramendu Bhattacharjee
2008-09-01
The stretching and bending vibrational energies of SnBr4 and CCl4 are calculated in the one-dimensional framework. The dynamical symmetry group of tetrahedral molecule was taken into consideration to construct the model Hamiltonian in this frame-work. Casimir and Majorana invariant operators were also determined accordingly. Using the model Hamiltonian so constructed, we reported the vibrational energy levels of SnBr4 and CCl4 molecules accurately.
2 Algebra and two-dimensional quasiexactly solvable Hamiltonian related to Poschl–Teller potential
Indian Academy of Sciences (India)
H Panahi; H Rahmati
2014-07-01
In this article, we write the general form of the quasiexactly solvable Hamiltonian of 2 algebra via one special representation in the – two-dimensional space. Then, by choosing an appropriate set of coefficients and making a gauge rotation, we show that this Hamiltonian leads to the solvable Poschl–Teller model on an open infinite strip. Finally, we assign 2 hidden algebra to the Poschl–Teller potential and obtain its spectrum by using the representation space of 2 algebra.
Almost-graded central extensions of Lax operator algebra
Schlichenmaier, Martin
2011-01-01
Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for $\\gl(n)$, with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. These results ...
Directory of Open Access Journals (Sweden)
Amira Ghorbel
2009-02-01
Full Text Available The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if G = N × A is a connected, simply connected, nilpotent Lie group with an Abelian factor A, then every uniform subgroup of G is the direct product of a uniform subgroup of N and Z^r where r = dim A.
Infinite-Dimensional Lie Superalgebras SHO' over a Field of Prime Characteristic
Institute of Scientific and Technical Information of China (English)
HE Ying-hua; YANG Xi-geng; LI Yu-xia
2008-01-01
The natural filtrations of the infinite-dimensional modular Lie superalgebra SHO' are proved to be invariant under automorphisms of SHO'. The proof involves the investigation of the ad-nilpotent elements of the even part, and the determination of the subalgebras generated by certain ad-nilpotent elements. A property of automorphisms of these Lie superalgebras can be established, and an intrinsic characterization of SHO' can be obtained.
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.
D=3(p,q)-Poincare supergravities from Lie algebra expansions
Energy Technology Data Exchange (ETDEWEB)
Azcarraga, J.A. de, E-mail: j.a.de.azcarraga@ific.uv.es [Dept. Theor. Phys. and IFIC (CSIC-UVEG), Univ. of Valencia, 46100-Burjassot (Valencia) (Spain); Izquierdo, J.M. [Dept. Theor. Phys., Univ. of Valladolid, 47011-Valladolid (Spain)
2012-01-01
We use the expansion of superalgebras procedure (summarized in the text) to derive Chern-Simons (CS) actions for the (p,q)-Poincare supergravities in three-dimensional spacetimes. After deriving the action for the (p,0)-Poincare supergravity as a CS theory for the expansion osp(p|2;R)(2,1) of osp(p|2;R), we find the general (p,q)-Poincare superalgebras and their associated D=3 supergravity actions as CS gauge theories from an expansion of the simple osp(p+q|2,R) superalgebras, namely osp(p+q|2,R)(2,1,2).
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。
The explicit structure of the nonlinear Schrödinger prolongation algebra
Eck, van H.N.; Gragert, P.K.H.; Martini, R.
1983-01-01
The structure of the nonlinear Schrödinger prolongation algebra, introduced by Estabrook and Wahlquist, is explicitly determined. It is proved that this Lie algebra is isomorphic with the direct product H× (A1 C[t]), where H is a three-dimensional commutative Lie algebra.
Oostveen, J
1996-01-01
In this paper we present results about the algebraic Riccati equation (ARE) and a weaker version of the ARE, the algebraic Riccati system (ARS), for infinite-dimensional, discrete-time systems. We introduce an operator pencil, associated with these equations, the so-called extended symplectic Pencil
The homoclinic and heteroclinic C*-algebra of a generalized one-dimensional solenoid
DEFF Research Database (Denmark)
Thomsen, Klaus
2010-01-01
D. Ruelle and I. Putnam have constructed three C*-algebras from the homoclinic and heteroclinic structure of a Smale space. This paper gives gives a complete description of these algebras when the Smale space is one of the generalized one-dimensional solenoids studied by R. Williams and I. Yi....
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.
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.
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.
Institute of Scientific and Technical Information of China (English)
Souleymanou Abbagari; Thomas B.Bouetou; Timoleon C.Kofane
2013-01-01
The prolongation structure methodologies of Wahlquist-Estabrook [H.D.Wahlquist and F.B.Estabrook,J.Math.Phys.16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system.Based on the obtained prolongation structure,a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed.A Lie-Algebra representation of some hidden structural symmetries of the previous system,its B(a)cklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived.In the wake of the previous results,we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + 1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation,which applications are straightforward in nonlinear optics where 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.
Classification of 3 -Lie bialgebras of type ( L b , C b
Directory of Open Access Journals (Sweden)
Bai Ruipu
2016-03-01
Full Text Available The 4-dimensional 3-Lie coalgebras with one-dimensional derived algebra, and 4-dimensional 3-Lie bialgebras of type (L_b, C_b are classified. It is proved that there exist two classes 4-dimensional 3-Lie coalgebras with one-dimensional derived algebra which are (L, C_{b_1} and (L, C_{b_2} (Theorem 3.2, and seven classes of 4-dimensional 3-Lie bialgebras of type (L_b, C_b (Theorem 3.3.
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
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2017-04-01
The representation theory of three dimensional real and complex Lie groups is reviewed from the perspective of harmonic functions defined over certain appropriate manifolds. An explicit construction of all unitary representations is given. The realisations obtained are shown to be related with each other by either natural operations as real forms or Inönü-Wigner contractions.
K1 Group of Finite Dimensional Path Algebra
Institute of Scientific and Technical Information of China (English)
Xue Jun GUO; Li Bin LI
2001-01-01
In this paper, by calculating the commutator subgroup of the unit group of finite pathalgebra κ/△ and the unit group abelianized, we explicitly characterize the K1 group of finite dimensionalpath algebra over an arbitrary field.
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.
Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld-Sokolov Hierarchy
Li, Chuan-Zhong; He, Jing-Song; Su, Yu-Cai
2014-04-01
In this paper, the dispersionless D-type Drinfeld-Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.
Institute of Scientific and Technical Information of China (English)
张玉峰
2003-01-01
A subalgebra of loop algebra A2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions,a formalism similar to the well-known Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and a generalized standard form of the Schrodinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra A2 into loop algebra A1. Furthermore,a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.
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
Low-lying Photoexcited States of a One-Dimensional Ionic Extended Hubbard Model
Yokoi, Kota; Maeshima, Nobuya; Hino, Ken-ichi
2017-10-01
We investigate the properties of low-lying photoexcited states of a one-dimensional (1D) ionic extended Hubbard model at half-filling. Numerical analysis by using the full and Lanczos diagonalization methods shows that, in the ionic phase, there exist low-lying photoexcited states below the charge transfer gap. As a result of comparison with numerical data for the 1D antiferromagnetic (AF) Heisenberg model, it was found that, for a small alternating potential Δ, these low-lying photoexcited states are spin excitations, which is consistent with a previous analytical study [Katsura et al., Phys. Rev. Lett. 103, 177402 (2009)]. As Δ increases, the spectral intensity of the 1D ionic extended Hubbard model rapidly deviates from that of the 1D AF Heisenberg model and it is clarified that this deviation is due to the neutral-ionic domain wall, an elementary excitation near the neutral-ionic transition point.
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））.
Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2-Dimensional Boundary Value Problems
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Roman Cherniha
2015-08-01
Full Text Available A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, followed as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1 + 2-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proven that there is a special exponent, k ≠ —2, for the power diffusivity uk when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case k ≠ —2. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity uk and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena to (1 + 1-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on the geometry of the domain are presented.
Schertzer, Daniel; Tchiguirinskaia, Ioula
2017-04-01
Multifractal fields have opened a new approach in geophysics to explore "spatial chaos", i.e. processes that are not only complex in time but also in space, because their definition is rather independent of their domain dimension. However multifractals have been for too long restricted to be scalar valued, i.e. to have one-dimensional codomains. This has prevented to deal with the key question of complex component interactions of vector fields and their non trivial symmetries. On the theoretical level, this is resolved by considering the Lie algebra of stochastic generators of cascade processes with arbitrarily large codomains, e.g. flows of vector fields over large dimensional manifolds. We recently investigated the neat example of stable Levy generators on Clifford algebra that provide both universal statistical and robust algebraic properties to the basic symmetries of the corresponding fields (Schertzer and Tchiguirinskaia, 2015). This presentation will focus on the concrete analysis of observation data and their simulation in the Levy-Clifford algebra framework. This correspond to a wide and innovative generalisation of classical multifractal methodologies. Schertzer, D. & Tchiguirinskaia, I., 2015. Multifractal vector fields and stochastic Clifford algebra. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(12), p.123127.
Randers metrics of Berwald type on four-dimensional hypercomplex Lie groups
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Moghaddam, H R Salimi [Department of Mathematics, Shahrood University of Technology, Shahrood (Iran, Islamic Republic of)], E-mail: hrsalimi@shahroodut.ac.ir
2009-03-06
In the present paper we study Randers metrics of Berwald type on simply connected four-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulae for computing the flag curvature of these metrics. By this study we construct two four-dimensional Berwald spaces, one of them with a non-negative flag curvature and the other one with a non-positive flag curvature.
Abstract Lie groups and locally compact topological groups
Directory of Open Access Journals (Sweden)
Jacek Lech
2004-05-01
Full Text Available We introduce a notion of abstract Lie group by means of the mapping which plays the role of the evolution operator. We show some basic properties of such groups very similar to the fundamentals of the infinite dimensional Lie theory. Next we give remarkable examples of abstract Lie groups which are not necessarily usual Lie groups. In particular, by making use of Yamabe theorem we prove that any locally compact topological group admits the structure of abstract Lie group and that the Lie algebra and the exponential mapping of it coincide with those determined by the Lie group structure.
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
Symmetries of a （2＋1）-Dimensional Toda-like Lattice
Institute of Scientific and Technical Information of China (English)
SHENShou-Feng; PANZu-Liang; ZHANGJun
2004-01-01
Lie group technique for solving differential-difference equations is applied to a new (2+1)-dimensional Toda-like lattice. An infinite dimensional Lie algebra and the corresponding commutation relations are obtained.
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.
k-Symplectic Lie systems: theory and applications
de Lucas, J.; Vilariño, S.
2015-03-01
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, t-independent constants of motion and general properties. Our results are illustrated through examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the k-symplectic geometry: systems of first-order ordinary differential equations.
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).
Non-Lie Symmetry Group and New Exact Solutions for the Two-Dimensional KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
WANG Hong; TIAN Ying-Hui; CHEN Han-Lin
2011-01-01
@@ By using the modified Clarkson-Kruskal (CK) direct method, we obtain the non-Lie symmetry group of the two-dimensional KdV-Burgers equation.Under some constraint conditions, Lie point symmetry is also obtained.Through the symmetry group, some new exact solutions of the two-dimensional KdV-Burgers equation are found.%By using the modified Clarkson-Kruskal (CK) direct method, we obtain the non-Lie symmetry group of the two-dimensional KdV-Burgers equation. Under some constraint conditions, Lie point symmetry is also obtained.Through the symmetry group, some new exact solutions of the two-dimensional KdV-Burgers equation are found.
Iftime, OV; Zwart, HJ; Curtain, RF
2005-01-01
We obtain a representation of all self-adjoint solutions of the control algebraic Riccati equation associated to the infinite-dimensional state linear system Sigma(A, B, C) under the following assumptions: A generates a C-0-group, the system is output stabilizable, strongly detectable and the dual R
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.
Inelastic light scattering by low-lying excitations of electrons in low-dimensional semiconductors
Energy Technology Data Exchange (ETDEWEB)
Pellegrini, V. [NEST CNR-INFM and Scuola Normale Superiore, Pisa (Italy); Pinczuk, A. [Department of Physics, Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027 (United States); Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey (United States)
2006-11-15
The low-dimensional electron systems that reside in artificial semiconductor heterostructures of great perfection are a contemporary materials base for explorations of collective phenomena. Studies of low-lying elementary excitations by inelastic light scattering offer insights on properties such energetics, interactions and spin magnetization. We review here recent light scattering results obtained from two-dimensional (2D) quantum fluids in semiconductor heterostructures under extreme conditions of low temperature and large magnetic field, where the quantum Hall phases are archetypes of novel behaviors. We also consider recent light scattering experiments that have probed the excitation spectra of few-electron states in semiconductor quantum dots. (copyright 2006 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
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.
Representations of a Class of Associative Algebras Related to the Quantum Torus
Institute of Scientific and Technical Information of China (English)
叶从峰
2003-01-01
@@ 1 Introduction The motivation of this paper comes from the work of [5]. We know that vertex algebra theory is one of the important parts in the study of infinite dimensional Lie theory, while the lattice vertex algebras[2,7] form one of the most important and fundamental classes of vertex algebras.
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.
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}.
On the Projective Algebra of Randers Metrics of Constant Flag Curvature
Directory of Open Access Journals (Sweden)
Mehdi Rafie-Rad
2011-08-01
Full Text Available The collection of all projective vector fields on a Finsler space (M,F is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by p(M,F and is the Lie algebra of the projective group P(M,F. The projective algebra p(M,F=α+β of a Randers space is characterized as a certain Lie subalgebra of the projective algebra p(M,α. Certain subgroups of the projective group P(M,F and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact n-manifold either equals n(n+2 or at most is n(n+1/2.
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.
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.
Algebraic classification of higher dimensional spacetimes based on null alignment
Ortaggio, Marcello; Pravdova, Alena
2012-01-01
We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some refinements and the generalized Newman-Penrose and Geroch-Held-Penrose formalisms. Next, we summarize general results, such as a partial extension of the Goldberg-Sachs theorem, characterization of spacetimes with vanishing (or constant) curvature invariants and the peeling behaviour in asymptotically flat spacetimes. Finally, we discuss certain invariantly defined families of metrics and their relation with the Weyl tensor classification, including: Kundt and Robinson-Trautman spacetimes; the Kerr-Schild ansatz in a constant-curvature background; purely electric and purely magnetic spacetimes; direct and (some) warped products; and geometries with certain symmetries. To conclude, some applications to quadratic gravity are also overviewed.
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.
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.
Orbifolds and Cosets of Minimal W-Algebras
Arakawa, Tomoyuki; Creutzig, Thomas; Kawasetsu, Kazuya; Linshaw, Andrew R.
2017-10-01
Let g be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of sl_2 inducing the minimal gradation on g. The corresponding minimal W-algebra W^k(g, e_{-θ})} introduced by Kac and Wakimoto has strong generators in weights {1,2,3/2}, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra {V^{k'}(g^{\
Jorgensen, PET
1987-01-01
Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas.This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly e
Diffeomorphism algebra of two dimensional free massless scalar field with signature change
Darabi, F; Rezaei-Aghdam, A
1999-01-01
We study a model of free massless scalar fields on a two dimensional cylinder with metric that admits a change of signature between Lorentzian and Euclidean type (ET), across the two timelike hypersurfaces (with respect to Lorentzian region). Considering a long strip-shaped region of the cylinder, denoted by an angle \\theta, as the signature changed region it is shown that the energy spectrum depends on the angle \\theta and in a sense differs from ordinary one for low energies. Morever diffeomorphism algebra of corresponding infinite conserved charges is different from '' Virasoro'' algebra and approaches to it at higher energies. The central term is also modified but does not approach to the ordinary one at higher energies.
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.
On some Lie groups as 5-dimensional almost contact B-metric manifolds with three natural connections
Ivanova, Miroslava; Manev, Hristo
2014-01-01
Almost contact manifolds with B-metric are considered. There are studied three natural connections (i.e. linear connections preserving the structure tensors) determined by conditions for their torsions. These connections are investigated on a family of Lie groups considered as 5-dimensional almost contact B-metric manifolds.
LIE SYMMETRY ANALYSIS AND PAINLEV(E) ANALYSIS OF THE NEW (2+1)-DIMENSIONAL KdV EQUATION
Institute of Scientific and Technical Information of China (English)
Shen Shoufeng
2007-01-01
Lie point symmetries associated with the new (2+l)-dimensional KdV equation ut + 3uxuy + uxxy = 0 are investigated. Some similarity reductions are derived by solving the the soliton solution is obtained directly from the B(a)cklund transformation.
CSIR Research Space (South Africa)
Fisher, JT
2015-07-01
Full Text Available Vegetation Science What lies beneath: detecting sub-canopy changes in savanna woodlands using a three-dimensional classification method Jolene T. Fisher, Ed T.F. Witkowski, Barend F.N. Erasmus, Penelope J. Mograbi, Gregory P. Asner, Jan A.N. van...
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.
Energy Technology Data Exchange (ETDEWEB)
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.
Polynomial algebras Poisson with regular structure of simplectical leaf
Odesskij, A V
2002-01-01
The Poisson polynomial algebras with certain regularity conditions are studied. The linear structure on the dual spaces of the semisimple Lie algebras, the Sklyanin quadratic elliptical algebras as well as the polynomial algebras constitute, in particular, the algebras of this class. Simple determinate relations between the brackets and Kazimir operators are established. These relations determine, in particular, that the sum of the Kazimir operators grades coincides with the dimensionality of the algebra for the Sklyanin elliptical algebras. The examples of such algebras are presented and it is shown, that some of them are naturally generated in the Hamiltonian integrated systems. The new class of the two-particle integrable systems, depending elliptically both on the coordinates and pulses, is among these examples
Pro-Lie Groups: A Survey with Open Problems
Directory of Open Access Journals (Sweden)
Karl H. Hofmann
2015-07-01
Full Text Available A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups. The article also lists 12 open questions connected to pro-Lie groups.
Institute of Scientific and Technical Information of China (English)
Ran SHEN; Yu Cai SU
2007-01-01
We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series).
Low-Lying States of the A+B-A+B- Coulomb Systems in Two-Dimensional Quantum Dots
Institute of Scientific and Technical Information of China (English)
XIE Wen-Fang
2001-01-01
The features of the low-lying spectra of four-body A+B-A+B- systems have been deduced based on symmetry. Using the method of few-body physics, we calculate the energy spectra of A + B- A + B- systems in a harmonic quantum dot. We find that the biexciton in a two-dimensional quantum dot may have other bound excited states and the quantum mechanical symmetry plays a crucialrole in determining the energy levels and structures of the low-lying states.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
A generalized variable-coefficient algebraic method is applied to construct several new families of exact solutions of physical interestfor (3+1)-dimensional Kadomtsev-Petviashvilli (KP) equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.
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.
L∞-algebra models and higher Chern-Simons theories
Ritter, Patricia; Sämann, Christian
2016-10-01
We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of L∞-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie p-algebra extensions of 𝔰𝔬(p + 2). Finally, we study a number of L∞-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.
Irreducible decomposition for tensor prodect representations of Jordanian quantum algebras
Aizawa, N
1997-01-01
Tensor products of irreducible representations of the Jordanian quantum algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest weight finite dimensional representations of U_h(sl(2)) and lowest weight infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding Lie algebras.
Unified (p,q;α,γ,l)-deformation of oscillator algebra and two-dimensional conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Burban, I.M., E-mail: burban@bitp.kiev.ua
2013-11-29
The unified (p,q;α,γ,l)-deformation of a number of well-known deformed oscillator algebras is introduced. The deformation is constructed by imputing new free parameters into the structure functions and by generalizing the defining relations of these algebras. The generalized Jordan–Schwinger and Holstein–Primakoff realizations of the U{sub pq}{sup αγl}(su(2)) algebra by the generalized (p,q;α,γ,l)-deformed operators are found. The generalized (p,q;α,γ,l)-deformation of the two-dimensional conformal field theory is established. By introducing the (p,q;α,γ,l)-operator product expansion (OPE) between the energy–momentum tensor and primary fields, we obtain the (p,q;α,γ,l)-deformed centerless Virasoro algebra. The two-point correlation function of the primary generalized (p,q;α,γ,l)-deformed fields is calculated.
Lie group classification and exact solutions of the generalized Kompaneets equations
Directory of Open Access Journals (Sweden)
Oleksii Patsiuk
2015-04-01
Full Text Available We study generalized Kompaneets equations (GKEs with one functional parameter, and using the Lie-Ovsiannikov algorithm, we carried out the group classification. It is shown that the kernel algebra of the full groups of the GKEs is the one-dimensional Lie algebra. Using the direct method, we find the equivalence group. We obtain six non-equivalent (up to transformations from the equivalence group GKEs that allow wider invariance algebras than the kernel one. We find a number of exact solutions of the non-linear GKE which has the maximal symmetry properties.
The $SW(3/2,2)$ superconformal algebra via a Quantum Hamiltonian Reduction of $osp(3|2)$
Díaz, Lázaro O Rodríguez
2016-01-01
We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which are extensions of the $N=1$ superconformal algebra by a primary supercurrent of conformal weight $2$ can be realized as a quantum Hamiltonian reduction of the Lie superalgebra $osp(3|2)$. In consequence we obtain an explicit free field realization of the algebra in terms of the screening operators. At central charge $c=12$ the $SW(3/2,2)$ superconformal algebra corresponds to the superconformal algebra associated to sigma models based on eight-dimensional manifolds with special holonomy $Spin(7)$, i.e., the Shatashvili-Vafa $Spin(7)$ superconformal algebra.
Non-integral representation theory of even multiplicity finite W-algebras
Brown, Jonathan S
2011-01-01
We complete the classification of the finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent elements in classical Lie algebras. This extends earlier work where this classification is determined for such representations of integral central character.
(2+1)-dimensional supersymmetric integrable equations
Yan, Zhao-Wen; Tala; Chen, Fang; Liu, Tao-Ran; Han, Jing-Min
2017-09-01
By means of two different approaches, we construct the (2+1)-dimensional supersymmetric integrable equations based on the super Lie algebra osp(3/2). We relax the constraint condition of homogenous space of super Lie algebra osp(3/2) in the first approach. In another one, the technique of extending the dimension of the systems is used. Furthermore for the (2 + 1)-dimensional supersymmetric integrable equations, we also derive their Bäcklund transformations.
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.
Orbifold Riemann surfaces: Teichmueller spaces and algebras of geodesic functions
Energy Technology Data Exchange (ETDEWEB)
Mazzocco, Marta [Loughborough University, Loughborough (United Kingdom); Chekhov, Leonid O [Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow (Russian Federation)
2009-12-31
A fat graph description is given for Teichmueller spaces of Riemann surfaces with holes and with Z{sub 2}- and Z{sub 3}-orbifold points (conical singularities) in the Poincare uniformization. The corresponding mapping class group transformations are presented, geodesic functions are constructed, and the Poisson structure is introduced. The resulting Poisson algebras are then quantized. In the particular cases of surfaces with n Z{sub 2}-orbifold points and with one and two holes, the respective algebras A{sub n} and D{sub n} of geodesic functions (classical and quantum) are obtained. The infinite-dimensional Poisson algebra D{sub n}, which is the semiclassical limit of the twisted q-Yangian algebra Y'{sub q}(o{sub n}) for the orthogonal Lie algebra o{sub n}, is associated with the algebra of geodesic functions on an annulus with n Z{sub 2}-orbifold points, and the braid group action on this algebra is found. From this result the braid group actions are constructed on the finite-dimensional reductions of this algebra: the p-level reduction and the algebra D{sub n}. The central elements for these reductions are found. Also, the algebra D{sub n} is interpreted as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point. Bibliography: 36 titles.
Numerical and algebraic studies for the control of finite-dimensional quantum systems
Energy Technology Data Exchange (ETDEWEB)
Sander, Uwe
2010-11-18
In this thesis, two aspects of control theory, namely controllability and optimal control, are applied to quantum systems. The presented results are based on group theoretical techniques and numerical studies. By Lie-algebraic analysis, the controllability properties of systems with an arbitrary topology are described and related to the symmetries existing in these systems. We find that symmetry precludes full controllability. Our work investigates well-known control systems and gives rules for the design of new systems. Furthermore, theoretical and numerical concepts are instrumental to studying quantum channels: Their capacities are optimised using gradient flows on the unitary group in order to find counterexamples to a long-established additivity conjecture. The last part of this thesis presents and benchmarks a modular optimal control algorithm known as GRAPE. Numerical tests show how the interplay of its modules can be optimised for higher performance, and how the algorithm performs in comparison to a Krotov-type optimal control algorithm. It is found that GRAPE performs particularly well when aiming for high qualities. (orig.)
Nishiyama, Yoshihiro
2017-03-01
The (2 + 1)-dimensional q = 3 Potts model was simulated with the exact diagonalization method. In the ordered phase, the elementary excitations (magnons) are attractive, forming a series of bound states in the low-energy spectrum. We investigate the low-lying spectrum through a dynamical susceptibility, which is readily tractable with the exact diagonalization method via the continued-fraction expansion. As a result, we estimate the series of (scaled) mass gaps, m 2 , 3 , 4 /m1 (m1: single-magnon mass), in proximity to the transition point.
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.
Lie symmetry group of (2+1-dimensional Jaulent-Miodek equation
Directory of Open Access Journals (Sweden)
Ma Hong-Cai
2014-01-01
Full Text Available In this paper, we consider a system of (2+1-dimensional non-linear model by using auxiliary equation method and Clarkson-Kruskal direct method which is very important in fluid and physics. We construct some new exact solutions of (2+1-dimensional non-linear models with the aid of symbolic computation which can illustrate some actions in fluid in the future.
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.
The Schroedinger-Virasoro algebra. Mathematical structure and dynamical Schroedinger symmetries
Energy Technology Data Exchange (ETDEWEB)
Unterberger, Jeremie [Henri Poincare Univ., Vandoeuvre-les-Nancy (France). Inst. Elie Cartan; Roger, Claude [Lyon I Univ., Villeurbanne (France). Dept. de Mathematiques
2012-07-01
This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure the Schroedinger-Virasoro algebra. Just as Poincare invariance or conformal (Virasoro) invariance play a key role in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence. The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schroedinger operators. (orig.)
The Schrödinger-Virasoro Algebra Mathematical structure and dynamical Schrödinger symmetries
Unterberger, Jérémie
2012-01-01
This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure—the Schrödinger-Virasoro algebra. Just as Poincaré invariance or conformal (Virasoro) invariance play a key role in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence. The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schrödinger operators. .