Sample records for algebra and number theory

  1. Algebra and Number Theory An Integrated Approach

    Dixon, Martyn; Subbotin, Igor


    Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines-linear algebra, abstract algebra, and number theory-into one compr

  2. Algebraic theory of numbers

    Samuel, Pierre


    Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal

  3. Algebraic number theory

    Jarvis, Frazer


    The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the fi...

  4. Algebraic number theory

    Weiss, Edwin


    Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te

  5. The theory of algebraic numbers

    Pollard, Harry


    An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.

  6. Classical theory of algebraic numbers

    Ribenboim, Paulo


    Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields Part One is devoted to residue classes and quadratic residues In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, iner...

  7. Distribution theory of algebraic numbers

    Yang, Chung-Chun


    The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth''s theorem Subspace theorems Vojta''s conjectures L-functions.

  8. Algebraic number theory and code design for Rayleigh fading channels

    Oggier, F


    Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available. The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutoria

  9. Cryptography by means of linear algebra and number theory

    Elfadel, Ajaeb


    ABSTRACT: This thesis focuses on the techniques of cryptography in linear algebra and number theory. We first give the necessary review on modular arithmetic. Under Linear Algebra, Hill cipher cryptographic technique and its variations are studied. Under number theory, on the other hand, the definition of Euler function, and some important theorems in this regard are given. The cryptographic techniques such as the Caesar cipher, Exponential transformations and the Public key cryptographic tec...

  10. Partial Fractions in Calculus, Number Theory, and Algebra

    Yackel, C. A.; Denny, J. K.


    This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.

  11. Notes on the Theory of Algebraic Numbers

    Wright, Steve


    A series of lecture notes on the elementary theory of algebraic numbers, using only knowledge of a first-semester graduate course in algebra (primarily groups and rings). No prerequisite knowledge of fields is required. Based primarily on the texts of E. Hecke, Lectures on the Theory of Algebraic Numbers, Springer-Verlag, 1981 (English translation by G. Brauer and J. Goldman) and D. Marcus, Number Fields, Springer, 1977.

  12. Real Algebraic Number Theory I: Diophantine Approximation Groups

    Gendron, T. M.


    This is the first of three papers introducing a paradigm within which global algebraic number theory for the reals may be formulated so as to make possible the synthesis of algebraic and transcendental number theory into a coherent whole. We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of K-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geo...

  13. Baxter Algebras, Stirling Numbers and Partitions

    Guo, Li


    Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras.

  14. Elementary number theory an algebraic approach

    Bolker, Ethan D


    This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and the Fermat conjecture for exponents three and four. The text contains abundant numerical examples and a particularly helpful collection of exercises, many of which are small research problems requiring substantial study or outside reading. Some problems call for new proofs for theorems already covered or for inductive explorations and proofs of theorems found in later chapters.Ethan D. Bolke

  15. Tilting theory and cluster algebras

    Reiten, Idun


    We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.

  16. Algebraic K-theory and algebraic topology

    This contribution treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers

  17. Characterizing the Development of Specialized Mathematical Content Knowledge for Teaching in Algebraic Reasoning and Number Theory

    Bair, Sherry L.; Rich, Beverly S.


    This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two…

  18. Algebraic theta functions and Eisenstein-Kronecker numbers

    Bannai, Kenichi; Kobayashi, Shinichi


    In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and $p$-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.

  19. Universal Algebras of Hurwitz Numbers

    A. Mironov; Morozov, A; Natanzon, S.


    Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.

  20. On the Theory of Algebraic Numbers with Elements of Small Height

    Göral, Haydar


    In this paper, we study the field of algebraic numbers with elements of small height.We first show that the nonstandard algebraic numbers whose logarithmic height isinfinitesimal has the Mann property, and exploiting this we obtain a diophantine approximation result in number theory.Then, we prove that the theory of algebraic numbers with elements of small height is not simple and has the independence property.We also relate the simplicity of a certain pair with Lehmer's conjecture and obtain...

  1. Problems and proofs in numbers and algebra

    Millman, Richard S; Kahn, Eric Brendan


    Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets in-depth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications in-depth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles. The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to “prove” or “sol...

  2. Algebraic and structural automata theory

    Mikolajczak, B


    Automata Theory is part of computability theory which covers problems in computer systems, software, activity of nervous systems (neural networks), and processes of live organisms development.The result of over ten years of research, this book presents work in the following areas of Automata Theory: automata morphisms, time-varying automata, automata realizations and relationships between automata and semigroups.Aimed at those working in discrete mathematics and computer science, parts of the book are suitable for use in graduate courses in computer science, electronics, telecommunications, and control engineering. It is assumed that the reader is familiar with the basic concepts of algebra and graph theory.

  3. Lectures on algebraic quantum field theory and operator algebras

    In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as why mathematicians are/should be interested in algebraic quantum field theory would be equally fitting. besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to frontier research problems in mathematical. (author)

  4. Lectures on algebraic quantum field theory and operator algebras

    Schroer, Bert [Berlin Univ. (Germany). Institut fuer Theoretische Physik. E-mail:


    In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as why mathematicians are/should be interested in algebraic quantum field theory would be equally fitting. besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to frontier research problems in mathematical. (author)

  5. Certain number-theoretic episodes in algebra

    Sivaramakrishnan, R


    Many basic ideas of algebra and number theory intertwine, making it ideal to explore both at the same time. Certain Number-Theoretic Episodes in Algebra focuses on some important aspects of interconnections between number theory and commutative algebra. Using a pedagogical approach, the author presents the conceptual foundations of commutative algebra arising from number theory. Self-contained, the book examines situations where explicit algebraic analogues of theorems of number theory are available. Coverage is divided into four parts, beginning with elements of number theory and algebra such as theorems of Euler, Fermat, and Lagrange, Euclidean domains, and finite groups. In the second part, the book details ordered fields, fields with valuation, and other algebraic structures. This is followed by a review of fundamentals of algebraic number theory in the third part. The final part explores links with ring theory, finite dimensional algebras, and the Goldbach problem.

  6. Theory of The Generalized Bernoulli-Hurwitz Numbers for The Algebraic Functions of Cyclotomic Type and The Universal Bernoulli Numbers

    Ônishi, Yoshihiro


    Hurwitz numbers are the Laurent coefficients of an elliptic function $\\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher genus cases. They satisfy completely von Staudt-Clausen type theorem, an extension of von Staudt second theorem, and Kummer type congruence relation. The present paper is revised and combined version of math.NT/0304377 and math.NT/0312178 containing many nu...

  7. Number theory and its history

    Ore, Oystein


    A prominent mathematician presents the principal ideas and methods of number theory within a historical and cultural framework. Oystein Ore's fascinating, accessible treatment requires only a basic knowledge of algebra. Topics include prime numbers, the Aliquot parts, linear indeterminate problems, congruences, Euler's theorem, classical construction problems, and many other subjects.

  8. Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras

    Mahanta, Snigdhayan


    We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.

  9. Kleene Algebra with Products and Iteration Theories

    Kozen, Dexter; Mamouras, Konstantinos


    We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism.

  10. Fourier theory and C∗-algebras

    Bédos, Erik; Conti, Roberto


    We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.

  11. Scaling algebras and renormalization group in algebraic quantum field theory

    For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary spacetime manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined. (orig.)

  12. Representation Theory of Algebraic Groups and Quantum Groups

    Gyoja, A; Shinoda, K-I; Shoji, T; Tanisaki, Toshiyuki


    Invited articles by top notch expertsFocus is on topics in representation theory of algebraic groups and quantum groupsOf interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics

  13. C*-algebras and operator theory

    Murphy, Gerald J


    This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.

  14. Excision in algebraic K-theory and Karoubi's conjecture.

    Suslin, A A; Wodzicki, M


    We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130

  15. Topics in algebraic and topological K-theory

    Baum, Paul Frank; Meyer, Ralf; Sánchez-García, Rubén; Schlichting, Marco; Toën, Bertrand


    This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

  16. L^2-Betti Numbers of Algebras and Equivalence Relations


    In this thesis, we consider the L^2-Betti numbers associated to algebras and equivalence relations. For algebras, we expand upon the motivation between Connes and Shlyakhtenko's definition of L^2-Betti numbers of tracial algebras, and give an alternative definition of these numbers through an excision-type theorem. For equivalence relations we consider the definitions given by Gaboriau and Sauer and show that they coincide, in the process proving a theorem of Gaboriau. The methods used are st...

  17. Galois theory, motives and transcendental numbers

    Andre, Yves


    From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $\\pi$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number ...

  18. Topological conformal algebra and BRST algebra in non-critical string theories

    The operator algebra in non-critical string theories is studied by treating the cosmological term as a perturbation. The algebra of covariantly regularized BRST and related currents contains a twisted N = 2 superconformal algebra only at d = -2 in bosonic strings, and a twisted N = 3 superconformal algebra only at d = ±∞ in spinning strings. The bosonic string at d = -2 is examined by replacing the string coordinate by a fermionic matter with c = -2. The resulting bc-βγ system accommodates various forms of BRST cohomology, and the ghost number assignment and BRST cohomology are different in the c = -2 string theory and two-dimensional topological gravity. (author)

  19. An interface between physics and number theory

    Duchamp, Gérard Henry Edmond; Solomon, Allan I; Goodenough, Silvia


    We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate {\\it inter alia} a basis for concluding that the Euler gamma constant $\\gamma$ may be rational.

  20. Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers

    Zheng, Quan


    We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.

  1. Algebraic Theories and (Infinity,1)-Categories

    Cranch, James


    We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central example, treated at length, is the theory of E_infinity spaces: this has a tidy combinatorial description in terms of span diagrams of finite sets. We introduce a theory of distributive laws, allowing us to describe objects with two distributing E_infinity stuctures. From this we produce a theory of E_infinity ring spaces. We also study grouplike objects, and produce theories modelling infinite loop spaces (or connective spectra), and infinite loop spaces with coherent multiplicative structure (or connective ring spectra). We use this to construct the units of a grouplike E_infinity ring space in a natural manner. Lastly we provide a speculative pleasant description of the K-theory of monoidal quasicategories and quasicategories with ring-like structures.

  2. Nonassociativity, Malcev algebras and string theory

    Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality. (Copyright copyright 2013 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)

  3. Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

    Vehkalahti, Roope


    Since the invention of space-time coding numerous algebraic methods have been applied to code design. In particular algebraic number theory and central simple algebras have been at the forefront of the research. In the first part of the paper we will push this direction further and show how the error probability of algebraic codes is tied to some central aspects of algebraic number theory and central simple algebras. In particular we prove how the error probability of several algebraic codes is tied to the corresponding zeta functions and unit groups. In the second part of this paper we turn to study what information theory can say about algebra. We will first derive some corollaries from the diversity-multiplexing gain tradeoff (DMT) Zheng and Tse and later show how these results can be used to analyze the unit group of orders of certain division algebras.

  4. Quantum field theory and Hopf algebra cohomology

    Brouder, Christian [Laboratoire de Mineralogie-Cristallographie, CNRS UMR 7590, Universites Paris 6 et 7, IPGP, 4 Place Jussieu, F-75252 Paris Cedex 05 (France); Fauser, Bertfried [Universitaet Konstanz, Fachbereich Physik, Fach M678, D-78457 Konstanz (Germany); Frabetti, Alessandra [Institut Girard Desargues, CNRS UMR 5028, Universite de Lyon 1, 21 av. Claude Bernard, F-69622 Villeurbanne (France); Oeckl, Robert [Centre de Physique Theorique, CNRS UPR 7061, F-13288 Marseille Cedex 9 (France)


    We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulae for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.

  5. A Workshop on Algebraic Design Theory and Hadamard Matrices


    This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions. The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important ap...

  6. Conformal field theory, tensor categories and operator algebras

    This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or quantum field theory is assumed. (topical review)

  7. Introduction to algebraic independence theory

    Philippon, Patrice


    In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.

  8. Number theory

    Andrews, George E


    Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simpl

  9. Algebraic Graph Theory Morphisms, Monoids and Matrices

    Knauer, Ulrich


    This is a highly self-contained book about algebraic graph theory which iswritten with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures -like roads, computers, telephones -instances of abstract data structures -likelists, stacks, trees -and functional or object orient

  10. Duality and products in algebraic (co)homology theories

    Kowalzig, N.; Kraehmer, U.


    The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a ×A-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie–Rinehart (co)homology.

  11. Krichever-Novikov type algebras theory and applications

    Schlichenmaier, Martin


    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

  12. Deformation Quantization Observable Algebras, States and Representation Theory

    Waldmann, S


    In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star product algebra, physical applications require to study representations of this algebra. I review the recent development of a representation theory including techniques like Rieffel induction and Morita equivalence. Applications beyond quantization theory are found in noncommutative field theories.

  13. Integrable theories and generalized graded Maillet algebras

    We present a general formalism to investigate the integrable properties of a large class of non-ultralocal models which in principle allows the construction of the corresponding lattice versions. Our main motivation comes from the su(1|1) subsector of the string theory on AdS5 × S5 in the uniform gauge, where such type of non-ultralocality appears in the resulting Alday–Arutyunov–Frolov (AAF) model. We first show how to account for the second derivative of the delta function in the Lax algebra of the AAF model by modifying Maillet’s r- and s-matrices formalism, and derive a well-defined algebra of transition matrices, which allows for the lattice formulation of the theory. We illustrate our formalism on the examples of the bosonic Wadati–Konno–Ichikawa–Shimizu (WKIS) model and the two-dimensional free massive Dirac fermion model, which can be obtained by a consistent reduction of the full AAF model, and give the explicit forms of their corresponding r-matrices. (paper)

  14. Quantum groups and algebraic geometry in conformal field theory

    The classification of two-dimensional conformal field theories is described with algebraic geometry and group theory. This classification is necessary in a consistent formulation of a string theory. (author). 130 refs.; 4 figs.; schemes

  15. Cuntz-Krieger algebras and a generalization of Catalan numbers

    Matsumoto, Kengo


    We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$...

  16. International Conference on Semigroups, Algebras and Operator Theory

    Meakin, John; Rajan, A


    This book discusses recent developments in semigroup theory and its applications in areas such as operator algebras, operator approximations and category theory. All contributing authors are eminent researchers in their respective fields, from across the world. Their papers, presented at the 2014 International Conference on Semigroups, Algebras and Operator Theory in Cochin, India, focus on recent developments in semigroup theory and operator algebras. They highlight current research activities on the structure theory of semigroups as well as the role of semigroup theoretic approaches to other areas such as rings and algebras. The deliberations and discussions at the conference point to future research directions in these areas. This book presents 16 unpublished, high-quality and peer-reviewed research papers on areas such as structure theory of semigroups, decidability vs. undecidability of word problems, regular von Neumann algebras, operator theory and operator approximations. Interested researchers will f...

  17. Theory of Generalized Bernoulli-Hurwitz Numbers in the Algebraic Functions of Cyclotomic Type

    Ônishi, Yoshihiro


    In this paper we announce some results obtained for certain algebraic functions, which we call of cyclotomic type. The main results properly resemble von Staudt-Clausen's theorem and Kummer's congruence for the Bernoulli numbers, and such theorems for the Hurwitz numbers.

  18. Algebraic structure and Poisson's theory of mechanico-electrical systems

    Liu Hong-Ji; Tang Yi-Fa; Fu Jing-Li


    The algebraic structure and Poisson's integral theory of mechanico-electrical systems are studied.The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained.The Lie algebraic structure and the Poisson's integral theory of Lagrange mechanico-electrical systems are derived.The Lie algebraic structure admitted and Poisson's integral theory of the Lagrange-Maxwell mechanico-electrical systems are presented.Two examples are presented to illustrate these results.

  19. Australian Curriculum Linked Lessons: Reasoning in Number and Algebra

    Day, Lorraine


    The Reasoning Proficiency in number and algebra is about children making sense of the mathematics by explaining their thinking, giving reasons for their decisions and describing mathematical situations and concepts. Lorraine Day notes, children need to be able to speak, read and write the language of mathematics while investigating pattern and…

  20. Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory

    Marcolli, Matilde; Port, Alexander


    Graph grammars extend the theory of formal languages in order to model distributed parallelism in theoretical computer science. We show here that to certain classes of context-free and context-sensitive graph grammars one can associate a Lie algebra, whose structure is reminiscent of the insertion Lie algebras of quantum field theory. We also show that the Feynman graphs of quantum field theories are graph languages generated by a theory dependent graph grammar.

  1. Division Algebras, Supersymmetry and Higher Gauge Theory

    Huerta, John


    From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7 and 11. The existence of these cocycles follow from spinor identities that hold only in these dimensions, and which are closely related to the existence of the superstring in dimensions k+2, and the super-2-brane in dimensions k+3. In general, an (n+1)-cocycle on a Lie superalgebra yields a `Lie n-superalgebra': that is, roughly, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincare superalgebra in dimensions k+2, and Lie 3-superalgebras extending the Poincare superalgebra in dimensions k+3. We present evidence, based on the work of Sati, Schreiber and Stasheff, that these Lie n...

  2. Algebraic quantum field theory

    The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)

  3. Numerical algebra, matrix theory, differential-algebraic equations and control theory festschrift in honor of Volker Mehrmann

    Bollhöfer, Matthias; Kressner, Daniel; Mehl, Christian; Stykel, Tatjana


    This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on ...

  4. String field theory. Algebraic structure, deformation properties and superstrings

    This thesis discusses several aspects of string field theory. The first issue is bosonic open-closed string field theory and its associated algebraic structure - the quantum open-closed homotopy algebra. We describe the quantum open-closed homotopy algebra in the framework of homotopy involutive Lie bialgebras, as a morphism from the loop homotopy Lie algebra of closed string to the involutive Lie bialgebra on the Hochschild complex of open strings. The formulation of the classical/quantum open-closed homotopy algebra in terms of a morphism from the closed string algebra to the open string Hochschild complex reveals deformation properties of closed strings on open string field theory. In particular, we show that inequivalent classical open string field theories are parametrized by closed string backgrounds up to gauge transformations. At the quantum level the correspondence is obstructed, but for other realizations such as the topological string, a non-trivial correspondence persists. Furthermore, we proof the decomposition theorem for the loop homotopy Lie algebra of closed string field theory, which implies uniqueness of closed string field theory on a fixed conformal background. Second, the construction of string field theory can be rephrased in terms of operads. In particular, we show that the formulation of string field theory splits into two parts: The first part is based solely on the moduli space of world sheets and ensures that the perturbative string amplitudes are recovered via Feynman rules. The second part requires a choice of background and determines the real string field theory vertices. Each of these parts can be described equivalently as a morphism between appropriate cyclic and modular operads, at the classical and quantum level respectively. The algebraic structure of string field theory is then encoded in the composition of these two morphisms. Finally, we outline the construction of type II superstring field theory. Specific features of the

  5. Algebraic Signal Processing Theory

    Pueschel, Markus; Moura, Jose M. F.


    This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a linear signal model defined as a triple (A, M, phi), where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively, and phi generalizes the concept of the z-transform to bijective linear mappings from a vector space of, e.g., signal samples, into the module M. A signal model provides the structure for a p...

  6. Bipartite field theories, cluster algebras and the Grassmannian

    We review recent progress in bipartite field theories. We cover topics such as their gauge dynamics, emergence of toric Calabi–Yau manifolds as master and moduli spaces, string theory embedding, relationships to on-shell diagrams, connections to cluster algebras and the Grassmannian, and applications to graph equivalence and stratification of the Grassmannian. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’. (review)

  7. Algebraic Quantization, Good Operators and Fractional Quantum Numbers

    Aldaya Valverde, Víctor; Calixto Molina, Manuel; Guerrero García, Julio


    The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the “failure” of the Ehrenfest theorem is clarified in terms of the already defined notion of good (and bad) operators. The analysis of “constrained” Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric ...

  8. Operators and representation theory canonical models for algebras of operators arising in quantum mechanics

    Jorgensen, PET


    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

  9. Developments and retrospectives in Lie theory algebraic methods

    Penkov, Ivan; Wolf, Joseph


    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...

  10. Fusion algebras and characters of rational conformal field theories

    Eholzer, W


    We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our first main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. Secondly, we show that the conformal characters of various rational models of W-algebras can be determined from the mere knowledge of the central charge and the set of conformal dimensions. We also describe how to actually construct conformal characters by using theta series associated to certain lattices. On our way we develop several tools for studying representations of the modular group on spaces of ...

  11. Representation, simplification and display of fractional powers of rational numbers in computer algebra

    Rich, Albert D.; Stoutemyer, David R.


    Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail. Since they are such a restricted subset of algebraic numbers, it might seem that good simplification of them must already be implemented in all widely used computer algebra systems. However, the algorithm taught in beginning algebra uses integer factorization...

  12. Algebraic and combinatorial Brill-Noether theory

    Caporaso, Lucia


    The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for smooth projective curves. Similarly, the existence of a graph for which the Brill-Noether variety is empty implies the emptiness of the corresponding Brill-Noether variety for a general curve. The main tool is a refinement of Baker's Specialization Lemma.

  13. Contemporary developments in algebraic K-theory

    The School and Conference on Algebraic K-theory which took place at ICTP July 8-26, 2002 was a follow-up to the earlier one in 1997, and like its predecessor, the 2002 meeting endeavoured to emphasise the multidisciplinary aspects of the subject. However, one special feature of the 2002 School and Conference is that the whole activity was dedicated to H. Bass, one of the founders of Algebraic K-theory, on the occasion of his seventieth birthday. The School during the first two weeks, July 8 to 19 was devoted to expository lectures meant to explore and highlight connections between K-theory and several other areas of mathematics - Algebraic Topology, Number theory, Algebraic Geometry, Representation theory, and Non-commutative Geometry. This volume, constituting the Proceedings of the School, is dedicated to H. Bass. The Proceedings of the Conference during the last week July 22 - 26, which will appear in Special issues of K-theory, is also dedicated to H. Bass. The opening contribution by M. Karoubi to this volume consists of a comprehensive survey of developments in K-theory in the last forty-five years, and covers a very broad spectrum of the subject, including Topological K-theory, Atiyah-Singer index theorem, K-theory of Banach algebras, Higher Algebraic K-theory, Cyclic Homology etc. J. Berrick's contribution on 'Algebraic K-theory and Algebraic Topology' treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers. The contributions by M. Kolster titled 'K-theory and Arithmetics' includes such topics as values of zeta functions and relations to K-theory, K-theory of integers in number fields and associated conjectures, Etale cohomology, Iwasawa theory etc. A.O. Kuku's contributions on 'K-theory and Representation theory

  14. Kac-Moody algebras in gravity and M-theories

    Houart, Laurent


    The formulation of gravity and M-theories as very-extended Kac-Moody invariant theories is reviewed. Exact solutions describing intersecting extremal brane configurations smeared in all directions but one are presented. The intersection rules characterising these solutions are neatly encoded in the algebra. The existence of dualities for all G+++ and their group theoretical-origin are discussed.

  15. Wilson operator algebras and ground states for coupled BF theories

    Tiwari, Apoorv; Chen, Xiao; Ryu, Shinsei


    The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theori...

  16. Function algebras on finite sets basic course on many-valued logic and clone theory

    Lau, Dietlinde


    Gives an introduction to the theory of function algebras. This book gives the general concepts of the Universal Algebra in order to familiarize the reader from the beginning on with the algebraic side of function algebras. It is a source on function algebras for students and researchers in mathematical logic and theoretical computer science.

  17. Bicomplex holomorphic functions the algebra, geometry and analysis of bicomplex numbers

    Luna-Elizarrarás, M Elena; Struppa, Daniele C; Vajiac, Adrian


    The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers. Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. ...

  18. Valued Graphs and the Representation Theory of Lie Algebras

    Joel Lemay


    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.

  19. The gauge algebra of double field theory and Courant brackets

    We investigate the symmetry algebra of the recently proposed field theory on a doubled torus that describes closed string modes on a torus with both momentum and winding. The gauge parameters are constrained fields on the doubled space and transform as vectors under T-duality. The gauge algebra defines a T-duality covariant bracket. For the case in which the parameters and fields are T-dual to ones that have momentum but no winding, we find the gauge transformations to all orders and show that the gauge algebra reduces to one obtained by Siegel. We show that the bracket for such restricted parameters is the Courant bracket. We explain how these algebras are realised as symmetries despite the failure of the Jacobi identity.

  20. Topological insulators and C∗-algebras: Theory and numerical practice

    Hastings, Matthew B.; Loring, Terry A.


    We apply ideas from C∗-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to 12 3, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an "order parameter" for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C∗-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.

  1. Representation theory of current algebra and conformal field theory on Riemann surfaces

    We study conformal field theories with current algebra (WZW-model) on general Riemann surfaces based on the integrable representation theory of current algebra. The space of chiral conformal blocks defined as solutions of current and conformal Ward identities is shown to be finite dimensional and satisfies the factorization properties. (author)

  2. Kac-Moody Algebras and String Theory (THESIS)

    Cleaver, G B


    The focus of this thesis is on (1) the role of Ka\\v c-Moody (KM) algebras in string theory and the development of techniques for systematically building string theory models based on higher level ($K\\geq 2$) KM algebras and (2) fractional superstrings. In chapter two we review KM algebras and their role in string theory. In the next chapter, we present two results concerning the construction of modular invariant partition functions for conformal field theories built by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type $SU(2)_{K_A}xSU(2)_{K_B}$, finding all consistent theories for $K_A$ and $K_B$ odd. Second we show how known diagonal modular invariants can be used to construct inherently asymmetric invariants where the holomorphic and anti- hol...

  3. Gravity, Gauge Theories and Geometric Algebra

    Lasenby, Anthony; Doran, Chris; Gull, Stephen


    A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic' relations between physical fields. The properties of the gravitational gauge fields are derived from both classical ...

  4. Yang-Baxter algebras, integrable theories and Bethe Ansatz

    This paper presents the Yang-Baxter algebras (YBA) in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Behe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropriate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underlay the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized

  5. Fundamentals of number theory

    LeVeque, William J


    This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given - making the book self-contained in this respect.The author begins with an introductory chapter on number theory and its early history. Subsequent chapters deal with unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diopha

  6. Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions

    Buchstaber, Viktor M; Erokhovets, Nikolai Yu


    This survey is devoted to the classical problem of flag numbers of convex polytopes, and contains an exposition of results obtained on the basis of connections between the theory of convex polytopes and a number of modern directions of research. Bibliography: 62 titles.

  7. Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers

    R. Kannan; Lenstra, Arjen K.; Lovasz, L.


    It is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences, given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients. All algorithms work in polynomial time

  8. Unified theories for quarks and leptons based on Clifford algebras

    The general standpoint is presented that unified theories arise from gauging of Clifford algebras describing the internal degrees of freedom (charge, color, generation, spin) of the fundamental fermions. The general formalism is presented and the ensuing theories for color and charge (with extension to N colors), and for generations, are discussed. The possibility of further including the spin is discussed, also in connection with generations. (orig.)

  9. Decomposition numbers for Brauer algebras of type G(m,p,n) in characteristic zero

    Bowman, C.; Cox, A.


    We introduce Brauer algebras associated to complex reflection groups of type $G(m,p,n)$, and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.

  10. Brane Topological Field Theories and Hurwitz numbers for CW-complexes

    Natanzon, Sergey M.


    We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Froben...

  11. Exceptional supergravity theories, Jordan algebras and the magic square

    The Jordan formulation of quantum mechanics is equivalent to the ordinary Hilbert space formulation a la Dirac. The only exception being the Jordan algebra J/sub 3//sup 0/ of 3 x 3 hermitian octonionic matrices. The main motivation of Jordan in trying to generalize the algebraic framework of quantum mechanics was to be able to explain the new ''relativistic and nuclear phenomena'' that were observed at the time. In particular they had the β-decay phenomenon in mind. The unique possible generalization they found was considered to be ''too narrow for the generalization hoped for.'' Remarkably enough, fifty years after the work of JNW the exceptional Jordan algebra J/sub 3//sup 0/ has re-entered Physics in the framework of theories that attempt to unify all interactions. The authors refer to the exceptional supergravity theory. This theory or an extension thereof could provide us with a unique framework for a realistic unification of all interactions including gravity. If this is indeed the case then the early verdict of JNW on the exceptional Jordan algebra will have to be overturned and it will have its unique place in Physics as it has in Mathematics

  12. Arithmetic Deformation Theory of Lie Algebras

    Rastegar, Arash


    This paper is devoted to deformation theory of graded Lie algebras over $\\Z$ or $\\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian cat...

  13. Algebraic Graph Theory (a short course for postgraduate students and researchers)

    Shokrollahi, Arman


    Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. And the theory of association schemes and coherent configurations studies the algebra generated by associated matrices.

  14. Gravity, Gauge Theories and Geometric Algebra

    Lasenby, A; Gull, S F; Lasenby, Anthony; Doran, Chris; Gull, Stephen


    A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic' relations between physical fields. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure selects an action that is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These e...

  15. Elements of a theory of algebraic theories

    Hyland, Martin


    Kleisli bicategories are a natural environment in which the combinatorics involved in various notions of algebraic theory can be handled in a uniform way. The setting allows a clear account of comparisons between such notions. Algebraic theories, symmetric operads and nonsymmetric operads are treated as examples.

  16. Arithmetic geometry and number theory

    Weng, Lin


    Mathematics is very much a part of our culture; and this invaluable collection serves the purpose of developing the branches involved, popularizing the existing theories and guiding our future explorations.More precisely, the goal is to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works of Deninger-Werner in vector bundles on curves over p-adic fields; of Jiang on local gamma factors in automorphic representations; of Weng on Deligne pairings and Takhtajan-Zograf metrics; of Yoshida on CM-periods; of Yu on transcendence of specia

  17. Number Theory, Analysis and Geometry

    Goldfeld, Dorian; Jones, Peter


    Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang's vast contribution to mathematics, th

  18. Mathematics of the 19th century mathematical logic, algebra, number theory, probability theory

    Yushkevich, A


    This multi-authored effort, Mathematics of the nineteenth century (to be fol­ lowed by Mathematics of the twentieth century), is a sequel to the History of mathematics fram antiquity to the early nineteenth century, published in three 1 volumes from 1970 to 1972. For reasons explained below, our discussion of twentieth-century mathematics ends with the 1930s. Our general objectives are identical with those stated in the preface to the three-volume edition, i. e. , we consider the development of mathematics not simply as the process of perfecting concepts and techniques for studying real-world spatial forms and quantitative relationships but as a social process as weIl. Mathematical structures, once established, are capable of a certain degree of autonomous development. In the final analysis, however, such immanent mathematical evolution is conditioned by practical activity and is either self-directed or, as is most often the case, is determined by the needs of society. Proceeding from this premise, we intend...


    Giscard, P.-L; Rochet, P


    Trace monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and unicity of a prime factorization. Because of this, the set of hikes par...

  20. On logical, algebraic, and probabilistic aspects of fuzzy set theory

    Mesiar, Radko


    The book is a collection of contributions by leading experts, developed around traditional themes discussed at the annual Linz Seminars on Fuzzy Set Theory. The different chapters have been written by former PhD students, colleagues, co-authors and friends of Peter Klement, a leading researcher and the organizer of the Linz Seminars on Fuzzy Set Theory. The book also includes advanced findings on topics inspired by Klement’s research activities, concerning copulas, measures and integrals, as well as aggregation problems. Some of the chapters reflect personal views and controversial aspects of traditional topics, while others deal with deep mathematical theories, such as the algebraic and logical foundations of fuzzy set theory and fuzzy logic. Originally thought as an homage to Peter Klement, the book also represents an advanced reference guide to the mathematical theories related to fuzzy logic and fuzzy set theory with the potential to stimulate important discussions on new research directions in the fiel...

  1. Wilson operator algebras and ground states for coupled BF theories

    Tiwari, Apoorv; Ryu, Shinsei


    The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theories on the three-torus, we explicitly calculate the $\\mathcal{S}$- and $\\mathcal{T}$-matrices, which encode fractional braiding statistics and topological spin of loop-like excitations, respectively. In the coupled $BF$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with point-like excitations, form composite particles.

  2. Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory

    Vergu, C


    Scattering amplitudes in N = 4 super-Yang Mills theory can be computed to higher perturbative orders than in any other four-dimensional quantum field theory. The results are interesting transcendental functions. By a hidden symmetry (dual conformal symmetry) the arguments of these functions have a geometric interpretation in terms of configurations of points in CP^3 and they turn out to be cluster coordinates. We briefly introduce cluster algebras and discuss their Poisson structure and the Sklyanin bracket. Finally, we present a 40-term trilogarithm identity which was discovered by accident while studying the physical results.

  3. Valued Graphs and the Representation Theory of Lie Algebras

    Lemay, Joel


    Quivers (directed graphs) and 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 that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as "crushed" species...

  4. W-algebras in conformal field theory

    Quantum W-algebras are defined and their relevance for conformal field theories is outlined. We describe direct constructions of W-algebras using associativity requirements. With this approach one explicitly obtains the first members of series of W-algebras, including in particular 'Casimir algebras' (related to simple Lie algebras) and extended symmetry algebras corresponding to special Virasoro-minimal models. We also describe methods for the study of highest weight representations of W-algebras. In some cases consistency of the corresponding quantum field theory already imposes severe restrictions on the admitted representations, i.e. allows to determine the field content. We conclude by reviewing known results on W-algebras and RCFTs and show that most known rational conformal fields theories can be described in terms of Casimir algebras although on the level of W-algebras exotic phenomena occur. (author). 40 refs

  5. Clifford algebra, geometric algebra, and applications

    Lundholm, Douglas; Svensson, Lars


    These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The v...

  6. From rational numbers to algebra: separable contributions of decimal magnitude and relational understanding of fractions.

    DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J


    To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions. PMID:25744594

  7. Algebraic K-theory and abstract homotopy theory

    Blumberg, Andrew J


    We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature.

  8. Advanced number theory

    Cohn, Harvey


    ""A very stimulating book ... in a class by itself."" - American Mathematical MonthlyAdvanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.The book is divided into three parts

  9. Supersymmetry and DLCQ Limit of Lie 3-algebra Model of M-theory

    Sato, Matsuo


    In arXiv:1003.4694, we proposed two models of M-theory, Hermitian 3-algebra model and Lie 3-algebra model. In this paper, we study the Lie 3-algebra model with a Lorentzian Lie 3-algebra. This model is ghost-free despite the Lorentzian 3-algebra. We show that our model satisfies two criteria as a model of M-theory. First, we show that the model possesses N=1 supersymmetry in eleven dimensions. Second, we show the model reduces to BFSS matrix theory with finite size matrices in a DLCQ limit.

  10. Cryptography and computational number theory

    Shparlinski, Igor; Wang, Huaxiong; Xing, Chaoping; Workshop on Cryptography and Computational Number Theory, CCNT'99


    This volume contains the refereed proceedings of the Workshop on Cryptography and Computational Number Theory, CCNT'99, which has been held in Singapore during the week of November 22-26, 1999. The workshop was organized by the Centre for Systems Security of the Na­ tional University of Singapore. We gratefully acknowledge the financial support from the Singapore National Science and Technology Board under the grant num­ ber RP960668/M. The idea for this workshop grew out of the recognition of the recent, rapid development in various areas of cryptography and computational number the­ ory. The event followed the concept of the research programs at such well-known research institutions as the Newton Institute (UK), Oberwolfach and Dagstuhl (Germany), and Luminy (France). Accordingly, there were only invited lectures at the workshop with plenty of time for informal discussions. It was hoped and successfully achieved that the meeting would encourage and stimulate further research in information and computer s...

  11. Algebraic Multigrid for Disordered Systems and Lattice Gauge Theories

    Best, C


    The construction of multigrid operators for disordered linear lattice operators, in particular the fermion matrix in lattice gauge theories, by means of algebraic multigrid and block LU decomposition is discussed. In this formalism, the effective coarse-grid operator is obtained as the Schur complement of the original matrix. An optimal approximation to it is found by a numerical optimization procedure akin to Monte Carlo renormalization, resulting in a generalized (gauge-path dependent) stencil that is easily evaluated for a given disorder field. Applications to preconditioning and relaxation methods are investigated.

  12. Algebraic structures and eigenstates for integrable collective field theories

    Conditions for the construction of polynomial eigen-operators for the Hamiltonian of collective string field theories are explored. Such eigen-operators arise for only one monomial potential v(x)=μx2 in the collective field theory. They form a w∞-algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non-zero-energy polynomial eigen-operators. This analysis leads us to consider a particular potential ν(x)=μx2+g/x2. A Lie algebra of polynomial eigen-operators is then constructed for this potential. It is a symmetric 2-index Lie algebra, also represented as a subalgebra of U(sl(2)). (orig.)

  13. The role of difficulty and gender in numbers, algebra, geometry and mathematics achievement

    Rabab'h, Belal Sadiq Hamed; Veloo, Arsaythamby; Perumal, Selvan


    This study aims to identify the role of difficulty and gender in numbers, algebra, geometry and mathematics achievement among secondary schools students in Jordan. The respondent of the study were 337 students from eight public secondary school in Alkoura district by using stratified random sampling. The study comprised of 179 (53%) males and 158 (47%) females students. The mathematics test comprises of 30 items which has eight items for numbers, 14 items for algebra and eight items for geometry. Based on difficulties among male and female students, the findings showed that item 4 (fractions - 0.34) was most difficult for male students and item 6 (square roots - 0.39) for females in numbers. For the algebra, item 11 (inequality - 0.23) was most difficult for male students and item 6 (algebraic expressions - 0.35) for female students. In geometry, item 3 (reflection - 0.34) was most difficult for male students and item 8 (volume - 0.33) for female students. Based on gender differences, female students showed higher achievement in numbers and algebra compare to male students. On the other hand, there was no differences between male and female students achievement in geometry test. This study suggest that teachers need to give more attention on numbers and algebra when teaching mathematics.

  14. Elementary number theory

    Dudley, Underwood


    Ideal for a first course in number theory, this lively, engaging text requires only a familiarity with elementary algebra and the properties of real numbers. Author Underwood Dudley, who has written a series of popular mathematics books, maintains that the best way to learn mathematics is by solving problems. In keeping with this philosophy, the text includes nearly 1,000 exercises and problems-some computational and some classical, many original, and some with complete solutions. The opening chapters offer sound explanations of the basics of elementary number theory and develop the fundamenta

  15. On the binary expansions of algebraic numbers

    Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E.; Pomerance, Carl


    Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number {number_sign}(|y|, N) of 1-bits in the expansion of |y| through bit position N satisfies {number_sign}(|y|, N) > CN{sup 1/D} for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals {summation}{sub n{ge}0} 1/2{sup f(n)} where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we re-establish the transcendency of the Kempner--Mahler number {summation}{sub n{ge}0}1/2{sup 2{sup n}}, yet we can also handle numbers with a substantially denser occurrence of 1's. Though the number z = {summation}{sub n{ge}0}1/2{sup n{sup 2}} has too high a 1's density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z{sup 2}.

  16. Renormalization and periods in perturbative Algebraic Quantum Field Theory

    Rejzner, Kasia


    In this paper I give an overview of mathematical structures appearing in perturbative algebraic quantum field theory (pAQFT) and I show how these relate to certain periods. pAQFT is a mathematically rigorous framework that allows to build models of physically relevant quantum field theories on a large class of Lorentzian manifolds. The basic objects in this framework are functionals on the space of field configurations and renormalization method used is the Epstein-Glaser (EG) renormalization. The main idea in the EG approach is to reformulate the renormalization problem, using functional analytic tools, as a problem of extending almost homogeneously scaling distributions that are well defined outside some partial diagonals in $\\mathbb{R}^n$. Such an extension is not unique, but it gives rise to a unique "residue", understood as an obstruction for the extended distribution to scale almost homogeneously. Physically, such scaling violations are interpreted as contributions to the $\\beta$ function.

  17. Super Virasoro algebra and solvable supersymmetric quantum field theories

    Interesting and deep relationships between super Virasoro algebras and super soliton systems (super KdV, super mKdV and super sine-Gordon equations) are investigated at both classical and quantum levels. An infinite set of conserved quantities responsible for solvability is characterized by super Virasoro algebras only. Several members of the infinite set of conserved quantities are derived explicitly. (author)

  18. Higher algebraic K-theory an overview

    Lluis-Puebla, Emilio; Gillet, Henri; Soulé, Christophe; Snaith, Victor


    This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.

  19. Matrix algebra and sampling theory : The case of the Horvitz-Thompson estimator

    Dol, W.; Steerneman, A.G.M.; Wansbeek, T.J.


    Matrix algebra is a tool not commonly employed in sampling theory. The intention of this paper is to help change this situation by showing, in the context of the Horvitz-Thompson (HT) estimator, the convenience of the use of a number of matrix-algebra results. Sufficient conditions for the consisten

  20. New string theories and their generation number

    New heterotic string theories in four dimensions are constructed by tensoring a nonstandard SCFT along with some minimal SCFTs. All such theories are identified and their particle generation number is found. We prove that from the infinite number of new heterotic string theories only the {6} theory predicts three generations as seen in nature which makes it an interesting candidate for further study

  1. The Clifford algebra of physical space and Dirac theory

    Vaz, Jayme, Jr.


    The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.

  2. Algebraic and Topological Aspects of Rough Set Theory

    Vlach, Milan


    The main purpose of this talk is to show how some widely known and well established algebraic and topological notions are closely related to notions and results introduced and rediscovered in the rough set literature.

  3. A definition of graph homology and graph K-theory of algebras

    Movshev, M. V.


    We introduce and study elementary properties of graph homology of algebras. This new homology theory shares many features of cyclic and Hochschild homology. We also define a graph K-theory together with an analog of Chern character.

  4. Current algebra and conformal field theory on a figure eight

    Balachandran, A P; Sen-Gupta, K; Marmo, G; Salomonson, P; Simoni, A; Stern, A


    We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also fou...

  5. Schaum's outline of theory and problems of linear algebra

    Lipschutz, Seymour


    This third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.

  6. On the algebraic theory of kink sectors: Application to quantum field theory models and collision theory

    Several two dimensional quantum field theory models have more than one vacuum state. An investigation of super selection sectors in two dimensions from an axiomatic point of view suggests that there should be also states, called soliton or kink states, which interpolate different vacua. Familiar quantum field theory models, for which the existence of kink states have been proven, are the Sine-Gordon and the φ42-model. In order to establish the existence of kink states for a larger class of models, we investigate the following question: Which are sufficient conditions a pair of vacuum states has to fulfill, such that an interpolating kink state can be constructed? We discuss the problem in the framework of algebraic quantum field theory which includes, for example, the P(φ)2-models. We identify a large class of vacuum states, including the vacua of the P(φ)2-models, the Yukawa2-like models and special types of Wess-Zumino models, for which there is a natural way to construct an interpolating kink state. In two space-time dimensions, massive particle states are kink states. We apply the Haag-Ruelle collision theory to kink sectors in order to analyze the asymptotic scattering states. We show that for special configurations of n kinks the scattering states describe n freely moving non interacting particles. (orig.)

  7. Some C*-algebras associated to quantum gauge theories

    Hannabuss, Keith C.


    Algebras associated with Quantum Electrodynamics and other gauge theories share some mathematical features with T-duality Exploiting this different perspective and some category theory, the full algebra of fermions and bosons can be regarded as a braided Clifford algebra over a braided commutative boson algebra, sharing much of the structure of ordinary Clifford algebras.

  8. Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems.

    Geary, David C; Hoard, Mary K; Nugent, Lara; Rouder, Jeffrey N


    The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial-numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities. PMID:26255604

  9. Evolution algebras and their applications

    Tian, Jianjun Paul


    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.

  10. Algebraic K- and L-theory and applications to the topology of manifolds

    The development of geometric topology has led to the identification of specific algebraic structures of great richness and usefulness. A common theme in this area is the study of algebraic invariants of discrete groups or rings by topological methods. The resulting subject is now called algebraic K-theory. The purpose of these lecture notes is to survey some of the main constructions and techniques in algebraic K-theory, together with an indication of the topological background and applications. More details about proofs can be found in the references. The material is organized into some introductory sections, concerning linear and unitary K-theory, followed by descriptions of four important geometric problems and their related algebraic methods

  11. Experimental Number Theory, Part I : Tower Arithmetic

    Gnang, Edinah K.


    We introduce in this section an Algebraic and Combinatorial approach to the theory of Numbers. The approach rests on the observation that numbers can be identified with familiar combinatorial objects namely rooted trees, which we shall here refer to as towers. The bijection between numbers and towers provides some insights into unexpected connexions between Number theory, combinatorics and discrete probability theory.

  12. Algebraic equations an introduction to the theories of Lagrange and Galois

    Dehn, Edgar


    Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form. And they are given a more thorough exposition than is customary. For this reason, and also because the author concentrates on concrete applications of algebraic theory, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical ex

  13. Fierz identities for real Clifford algebras and the number of supercharges

    One considers supersymmetric gauge theories in quantum mechanics with the bosons and fermions belonging to the adjoint representation of the gauge group. One shows that the supersymmetry constraints are related to the existence of certain Fierz identities for real Clifford algebras. These identities are valid when one has 2, 4, 8, and 16 supercharges

  14. Conferences on Combinatorial and Additive Number Theory


    This proceedings volume is based on papers presented at the Workshops on Combinatorial and Additive Number Theory (CANT), which were held at the Graduate Center of the City University of New York in 2011 and 2012. The goal of the workshops is to survey recent progress in combinatorial number theory and related parts of mathematics. The workshop attracts researchers and students who discuss the state-of-the-art, open problems, and future challenges in number theory.

  15. Associative digital network theory an associative algebra approach to logic, arithmetic and state machines

    Benschop, Nico F


    ""Associative Digital Network Theory"" is intended for researchers at industrial laboratories, teachers and students at technical universities, in electrical engineering, computer science and applied mathematics departments, interested in new developments of modeling and designing digital networks (DN: state machines, sequential and combinational logic) in general, as a combined math/engineering discipline. As background an undergraduate level of modern applied algebra (Birkhoff-Bartee: ""Modern Applied Algebra"" - 1970, and Hartmanis-Stearns: ""Algebraic Structure of Sequential Machines"" - 1

  16. Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras

    Park, Jae-Suk


    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...

  17. Solutions in Bosonic String Field Theory and Higher Spin Algebras in AdS

    Polyakov, Dimitri


    We find a class of analytic solutions in open bosonic string field theory, parametrized by the chiral copy of higher spin algebra in $AdS_3$. The solutions are expressed in terms of the generating function for the products of Bell polynomials in derivatives of bosonic space-time coordinates $X^m(z)$ of the open string, which form is determined in this work. The products of these polynomials form a natural operator algebra realizations of $W_\\infty$ (area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin algebra in $AdS_3$. The class of SFT solutions found can, in turn, be interpreted as the "enveloping of enveloping", or the enveloping of $AdS_3$ higher spin algebra. We also discuss the extensions of this class of solutions to superstring theory and their relations to higher spin algebras in higher space-time dimensions.

  18. Abstract algebra structure and application

    Finston, David R


    This text seeks to generate interest in abstract algebra by introducing each new structure and topic via a real-world application. The down-to-earth presentation is accessible to a readership with no prior knowledge of abstract algebra. Students are led to algebraic concepts and questions in a natural way through their everyday experiences. Applications include: Identification numbers and modular arithmetic (linear) error-correcting codes, including cyclic codes ruler and compass constructions cryptography symmetry of patterns in the real plane Abstract Algebra: Structure and Application is suitable as a text for a first course on abstract algebra whose main purpose is to generate interest in the subject, or as a supplementary text for more advanced courses. The material paves the way to subsequent courses that further develop the theory of abstract algebra and will appeal to students of mathematics, mathematics education, computer science, and engineering interested in applications of algebraic concepts.

  19. Number theory arising from finite fields analytic and probabilistic theory

    Knopfmacher, John


    ""Number Theory Arising from Finite Fields: Analytic and Probabilistic Theory"" offers a discussion of the advances and developments in the field of number theory arising from finite fields. It emphasizes mean-value theorems of multiplicative functions, the theory of additive formulations, and the normal distribution of values from additive functions. The work explores calculations from classical stages to emerging discoveries in alternative abstract prime number theorems.

  20. Quantum field theory on toroidal topology: Algebraic structure and applications

    Khanna, F.C., E-mail: [Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2 (Canada); TRIUMF, Vancouver, BC, V6T 2A3 (Canada); Malbouisson, A.P.C., E-mail: [Centro Brasileiro de Pesquisas Físicas/MCT, 22290-180, Rio de Janeiro, RJ (Brazil); Malbouisson, J.M.C., E-mail: [Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA (Brazil); Santana, A.E., E-mail: [International Center for Condensed Matter Physics, Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF (Brazil)


    The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordström, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particle physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matter physics. The theory on a torus Γ{sub D}{sup d}=(S{sup 1}){sup d}×R{sup D−d} is developed from a Lie-group representation and c{sup ∗}-algebra formalisms. As a first application, the quantum field theory at finite temperature, in its real- and imaginary-time versions, is addressed by focusing on its topological structure, the torus Γ{sub 4}{sup 1}. The toroidal quantum-field theory provides the basis for a consistent approach of spontaneous symmetry breaking driven by both temperature and spatial boundaries. Then the superconductivity in films, wires and grains are analyzed, leading to some results that are comparable with experiments. The Casimir effect is studied taking the electromagnetic and Dirac fields on a torus. In this case, the method of analysis is based on a generalized Bogoliubov transformation, that separates the Green function into two parts: one is associated with the empty space–time, while the other describes the impact of compactification. This provides a natural procedure for calculating the renormalized energy–momentum tensor. Self interacting four-fermion systems, described by the Gross–Neveu and Nambu

  1. Factorization and selection rules of operator product algebras in conformal field theories

    Brustein, R.; Yankielowicz, S.; Zuber, J.B.


    Factorization of the operator product algebra in conformal field theory into independent left and right components is investigated. For those theories in which factorization holds we propose an ansatz for the number of independent amplitudes which appear in the fusion rules, in terms of the crossing matrices of conformal blocks in the plane. This is proved to be equivalent to a recent conjecture by Verlinde. The monodromy properties of the conformal blocks of 2-point functions on the torus are investigated. The analysis of their short-distance singularities leads to a precise definition of Verlinde's operations.

  2. Fractional Dirac operators and deformed field theory on Clifford algebra

    Fractional Dirac equations are constructed and fractional Dirac operators on Clifford algebra in four dimensional are introduced within the framework of the fractional calculus of variations recently introduced by the author. Many interesting consequences are revealed and discussed in some details.

  3. Open and Closed String field theory interpreted in classical Algebraic Topology

    Sullivan, Dennis


    There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.

  4. Teaching of Real Numbers by Using the Archimedes-Cantor Approach and Computer Algebra Systems

    Vorob'ev, Evgenii M.


    Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of…

  5. Factorisation and gauge transformations in supergravity theories constructed on free differential algebras

    Foussats, A.; Laura, R.; Zandron, O.


    Supergravity theories on a free differential algebra are examined and the factorisation condition is imposed leading to factorised solutions. H-gauge transformations for the pseudo-connections and pseudo-curvatures are also deduced.

  6. Factorisation and gauge transformations in supergravity theories constructed on free differential algebras

    Supergravity theories on a free differential algebra are examined and the factorisation condition is imposed leading to factorised solutions. H-gauge transformations for the pseudo-connections and pseudo-curvatures are also deduced. (author)

  7. Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras

    Davison, Ben


    This paper is a companion paper to 1512.08898, on the general definition of Donaldson--Thomas invariants for Jacobi algebras, or equivalently, the integrality conjecture for such algebras. In this paper we concentrate on the Hodge-theoretic aspects of the theory, and explore the structure of the Cohomological Hall algebra associated to a quiver and potential, introduced by Kontsevich and Soibelman. Via a study of the representation theory of these algebras, we introduce a perverse filtration on them, and prove that they are quantum enveloping algebras, for which the integrality theorem, and the wall crossing theorem relating DT invariants for different Bridgeland stability conditions, are a K-theoretic shadow of the existence of PBW bases.

  8. Enveloping -*-Algebra of a Smooth Frechet Algebra Crossed Product by $\\mathbb{R}, K$-Theory and Differential Structure in *-Algebras

    Subhash J Bhatt


    Given an -tempered strongly continuous action of $\\mathbb{R}$ by continuous $∗$-automorphisms of a Frechet $∗$-algebra , it is shown that the enveloping -*-algebra $E(S(\\mathbb{R},A^∞,))$ of the smooth Schwartz crossed product $S(\\mathbb{R},A^∞,)$ of the Frechet algebra $A^∞$ of $C^∞$-elements of is isomorphic to the -*-crossed product $C^∗(\\mathbb{R}, E(A), )$ of the enveloping -*-algebra () of by the induced action. When is a hermitian $\\mathcal{Q}$-algebra, one gets -theory isomorphism $R K_∗(S(\\mathbb{R},A^∞,))=K_∗(C^∗(\\mathbb{R}, E(A),)$ for the representable -theory of Frechet algebras. An application to the differential structure of a *-algebra defined by densely defined differential seminorms is given.

  9. The abc-conjecture for Algebraic Numbers

    Jerzy BROWKIN


    The abc-conjecture for the ring of integers states that, for every ε> 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1+ε with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.In the present paper we propose an abc-conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number.From this abc-conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over Q.

  10. The Poisson algebra of classical Hamiltonians in field theory and the problem of its quantization

    Stoyanovsky, A.


    We construct the commutative Poisson algebra of classical Hamiltonians in field theory. We pose the problem of quantization of this Poisson algebra. We also make some interesting computations in the known quadratic part of the quantum algebra.

  11. A Short History of the Theory of Numbers

    Alin Cristian Ioan


    The article treats some aspects of the history of the theory of algebraic numbers. The number theory is one of the oldest branches of mathematics which has its origins from second millennium BC, ancient documents dating from about 2000 BC being the Rhind papyrus and Golenischev papyrus, both of Egypt. Number theory is characterized by the simplicity of its fundamentals, its rigor and purity notions of its truths. One branch of number theory is the algebraic theory of numbers wh...

  12. Algebraic K-theory and derived equivalences suggested by T-duality for torus orientifolds

    Rosenberg, Jonathan


    We show that certain isomorphisms of (twisted) KR-groups that underlie T-dualities of torus orientifold string theories have purely algebraic analogues in terms of algebraic K-theory of real varieties and equivalences of derived categories of (twisted) coherent sheaves. The most interesting conclusion is a kind of Mukai duality in which the "dual abelian variety" to a smooth projective genus-1 curve over R with no real points is (mildly) noncommutative.

  13. Lie algebra cohomology and group structure of gauge theories

    We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincaracute e duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator Qdegree for the Lie algebra cohomology induced by BRST generator Q. We also point out an interesting duality relation emdash Poincaracute e duality emdash with respect to gauge anomalies and Wess endash Zumino endash Witten topological terms. We consider the consistent embedding of the BRST adjoint generator Qdegree into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint Noether charge Qdegree. copyright 1996 American Institute of Physics

  14. Lie algebra cohomology and group structure of gauge theories

    Yang, H.S.; Lee, B. [Department of Physics, Hanyang University, Seoul 133-791 (Korea)


    We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincar{acute e} duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator {ital Q}{sup {degree}} for the Lie algebra cohomology induced by BRST generator {ital Q}. We also point out an interesting duality relation{emdash}Poincar{acute e} duality{emdash}with respect to gauge anomalies and Wess{endash}Zumino{endash}Witten topological terms. We consider the consistent embedding of the BRST adjoint generator {ital Q}{sup {degree}} into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint N{umlt o}ther charge {ital Q}{sup {degree}}. {copyright} {ital 1996 American Institute of Physics.}

  15. KK-theory and Spectral Flow in von Neumann Algebras

    Kaad, Jens; Nest, Ryszard; Rennie, Adam


    We present a definition of spectral flow relative to any norm closed ideal J in any von Neumann algebra N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in K_0(J). In the case when N is semifinite, the numerical spectral flow of the...... path coincides with the value of trace on the associated K-class. Given a semifinite spectral triple (A,H,D) relative to a semifinite von Neumann algebra N, we construct a class [D] in KK^1(A,N') such that, for a unitary u in A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov...

  16. Number Theories

    St-Amant, Patrick


    We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization of the fundamental theorem of arithmetic for hyperoperations. We also give a generalized definition of the prime numbers that are associated to an arbitrary n-ary operation and take a few steps toward the development of its modulo arithmetic by investigating a generalized form of Fermat's little theorem. Those constructions give an interesting way to interpret diophantine equations and we will see that the uniqueness of factorization under an arbitrary operation can be linked with the Riemann zeta function. This language of generalized primes and composites can be used to restate and extend certain problems such as the Goldbach conjecture.

  17. Octonionic representations of Clifford Algebras and triality

    The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest Σ3x SO(8) structure in this framework

  18. Cylindric-like algebras and algebraic logic

    Ferenczi, Miklós; Németi, István


    Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways:  as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.

  19. Twisting theory for weak Hopf algebras

    CHEN Ju-zhen; ZHANG Yan; WANG Shuan-hong


    The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the (braided) monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl (2000).

  20. C-algebras and their applications to reflection groups and conformal field theories

    Zuber, Jean-Bernard


    The aim of this lecture is to present the concept of C-algebra and to illustrate its applications in two contexts: the study of reflection groups and their folding on the one hand, the structure of rational conformal field theories on the other. For simplicity the discussion is restricted to finite Coxeter groups and conformal theories with a $\\hat{sl}(2)$ current algebra, but it may be extended to a larger class of groups and theories associated with $\\hat{sl}(N)$. (Proceedings of the RIMS S...

  1. LieART -- A Mathematica Application for Lie Algebras and Representation Theory

    Feger, Robert


    We present the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie Algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. LieART can handle all classical and exceptional Lie algebras. It computes root systems of Lie algebras, weight systems and several other properties of irreducible representations. LieART's user interface has been created with a strong focus on usability and thus allows the input of irreducible representations via their dimensional name, while the output is in the textbook style used in most particle-physics publications. The unique Dynkin labels of irreducible representations are used internally and can also be used for input and output. LieART exploits the Weyl reflection group for most of the calculations, resulting in fast computations and a low memory consumption. Extensive tables of properties, tensor products and branching rules of irreducible ...

  2. Foliation theory in algebraic geometry

    McKernan, James; Pereira, Jorge


    Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013.  Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...

  3. Algebraic structure of cohomological field theory models and equivariant cohomology

    The definition of observables within conventional gauge theories is settled by general consensus. Within cohomological theories considered as gauge theories of an exotic type, that question has a much less obvious answer. It is shown here that in most cases these theories are best defined in terms of equivariant cohomologies both at the field level and at the level of observables. (author). 21 refs

  4. Algebraic structure of cohomological field theory models and equivariant cohomology

    Stora, R.; Thuillier, F. [Grenoble-1 Univ., 74 - Annecy (France). Lab. de Physique des Particules Elementaires; Wallet, J.Ch. [Paris-11 Univ., 91 - Orsay (France). Div. de Physique Theorique


    The definition of observables within conventional gauge theories is settled by general consensus. Within cohomological theories considered as gauge theories of an exotic type, that question has a much less obvious answer. It is shown here that in most cases these theories are best defined in terms of equivariant cohomologies both at the field level and at the level of observables. (author). 21 refs.

  5. Flux algebra, Bianchi identities and Freed-Witten anomalies in F-theory compactifications

    We discuss the structure of 4D gauged supergravity algebras corresponding to globally non-geometric compactifications of F-theory, admitting a local geometric description in terms of 10D supergravity. By starting with the well-known algebra of gauge generators associated to non-geometric type IIB fluxes, we derive a full algebra containing all, closed RR and NSNS, geometric and non-geometric dual fluxes. We achieve this generalization by a systematic application of SL(2,Z) duality transformations and by taking care of the spinorial structure of the fluxes. The resulting algebra encodes much information about the higher dimensional theory. In particular, tadpole equations and Bianchi identities are obtainable as Jacobi identities of the algebra. When a sector of magnetized (p,q) 7-branes is included, certain closed axions are gauged by the U(1) transformations on the branes. We indicate how the diagonal gauge generators of the branes can be incorporated into the full algebra, and show that Freed-Witten constraints and tadpole cancellation conditions for (p,q) 7-branes can be described as Jacobi identities satisfied by the algebra mixing bulk and brane gauge generators

  6. Infinite dimension algebra and conformal symmetry

    A generalisation of Kac-Moody algebras (current algebras defined on a circle) to algebras defined on a compact supermanifold of any dimension and with any number of supersymmetries is presented. For such a purpose, we compute all the central extensions of loop algebras defined on this supermanifold, i.e. all the cohomology classes of these loop algebras. Then, we try to extend the relation (i.e. semi-direct sum) that exists between the two dimensional conformal algebras (called Virasoro algebra) and the usual Kac-Moody algebras, by considering the derivation algebra of our extended Kac-Moody algebras. The case of superconformal algebras (used in superstrings theories) is treated, as well as the cases of area-preserving diffeomorphisms (used in membranes theories), and Krichever-Novikov algebras (used for interacting strings). Finally, we present some generalizations of the Sugawara construction to the cases of extended Kac-Moody algebras, and Kac-Moody of superalgebras. These constructions allow us to get new realizations of the Virasoro, and Ramond, Neveu-Schwarz algebras

  7. Linear algebra meets Lie algebra: the Kostant-Wallach theory

    Shomron, Noam; Parlett, Beresford N.


    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.

  8. Exploiting the Structure of Bipartite Graphs for Algebraic and Spectral Graph Theory Applications

    Kunegis, Jérôme


    In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering and commerce, many types of information can be represented as networks, and thus the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., representing graphs as square adjacency matrices. However, many networks are of a very specific form that...

  9. Hopf algebras and the combinatorics of connected graphs in quantum field theory

    Mestre, Angela; Oeckl, Robert


    In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in combinatorics or graph theory. It consists in finding a recursive algorithm that generates all connected graphs in their Hopf algebraic representation. This representation can be used directly and efficiently in evaluating Feynman graphs as contributions to the n-po...

  10. Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras

    Kodera, Ryosuke


    We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quantized Coulomb branch is a deformation of a subquotient of the Yangian of the affine $\\mathfrak{gl}(1)$.

  11. Value distribution theory of holomorphic curves into complex projective algebraic varieties and geometric diophantine problems

    We introduce a new technique transforming a holomorphic curve into a higher dimensional projective algebraic variety, f : C → X, to a system of holomorphic maps between appropriate Riemann surfaces, {λ : Yλ → Sλ}. Then we apply this transformation and its modifications to settle the conjectural Second Main Theorem in Nevanlinna theory for holomorphic curves into smooth complex projective algebraic varieties. Applications to geometric Diophantine problems are discussed. (author). 25 refs

  12. Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces

    Blumberg, Andrew J


    We interpret different constructions of algebraic $K$-theory of spaces as an instance of derived Koszul (or bar) duality and also as an instance of Morita equivalence. We relate the interplay between these two descriptions to the homotopy involution. We define a geometric analog of the Swan theory $G^{\\bZ}(\\bZ[\\pi])$ in terms of $\\Sigma^{\\infty}_{+} \\Omega X$ and show that it is the algebraic $K$-theory of the $E_{\\infty}$ ring spectrum $DX=S^{X_{+}}$.

  13. Higher Gauge Theories from Lie n-algebras and Off-Shell Covariantization

    Carow-Watamura, Ursula; Ikeda, Noriaki; Kaneko, Yukio; Watamura, Satoshi


    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 [1] (arxiv:1206.5643).

  14. Geometric Algebras and Extensors

    Fernandez, V. V.; Moya, A. M.; Rodrigues Jr., W. A.


    This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to met...

  15. Equivariant K-theory and freeness of group actions on C*-algebras

    Phillips, N Christopher


    Freeness of an action of a compact Lie group on a compact Hausdorff space is equivalent to a simple condition on the corresponding equivariant K-theory. This fact can be regarded as a theorem on actions on a commutative C*-algebra, namely the algebra of continuous complex-valued functions on the space. The successes of "noncommutative topology" suggest that one should try to generalize this result to actions on arbitrary C*-algebras. Lacking an appropriate definition of a free action on a C*-algebra, one is led instead to the study of actions satisfying conditions on equivariant K-theory - in the cases of spaces, simply freeness. The first third of this book is a detailed exposition of equivariant K-theory and KK-theory, assuming only a general knowledge of C*-algebras and some ordinary K-theory. It continues with the author's research on K-theoretic freeness of actions. It is shown that many properties of freeness generalize, while others do not, and that certain forms of K-theoretic freeness are related to ...

  16. Algebraic K-theory of generalized schemes

    Anevski, Stella Victoria Desiree

    Nikolai Durov has developed a generalization of conventional scheme theory in which commutative algebraic monads replace commutative unital rings as the basic algebraic objects. The resulting geometry is expressive enough to encompass conventional scheme theory, tropical algebraic geometry and...... geometry over the field with one element. It also permits the construction of important Arakelov theoretical objects, such as the completion \\Spec Z of Spec Z. In this thesis, we prove a projective bundle theorem for the eld with one element and compute the Chow rings of the generalized schemes Sp\\ec ZN...

  17. Operator algebras and topology

    These notes, based on three lectures on operator algebras and topology at the 'School on High Dimensional Manifold Theory' at the ICTP in Trieste, introduce a new set of tools to high dimensional manifold theory, namely techniques coming from the theory of operator algebras, in particular C*-algebras. These are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. A central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. This is an isomorphism conjecture, as discussed in the talks of Luck, but with a certain special flavor. Nevertheless, it has important direct applications to the topology of manifolds, it implies e.g. the Novikov conjecture. In the first chapter, the Baum-Connes conjecture will be explained and put into our context. Another application of the Baum-Connes conjecture is to the positive scalar curvature question. This will be discussed by Stephan Stolz. It implies the so-called 'stable Gromov-Lawson-Rosenberg conjecture'. The unstable version of this conjecture said that, given a closed spin manifold M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second chapter. The third chapter introduces another set of invariants, also using operator algebra techniques, namely L2-cohomology, L2-Betti numbers and other L2-invariants. These invariants, their basic properties, and the central questions about them, are introduced in the third chapter. (author)

  18. On the Central Charge of Spacetime Current Algebras and Correlators in String Theory on AdS3

    Kim, Jihun; Porrati, Massimo


    Spacetime Virasoro and affine Lie algebras for strings propagating in AdS3 are known to all orders in $\\alpha'$. The central extension of such algebras is a string vertex, whose expectation value can depend on the number of long strings present in the background but should be otherwise state-independent. In hep-th/0106004, on the other hand, a state-dependent expectation value was found. Another puzzling feature of the theory is lack of cluster decomposition property in certain connected corr...

  19. Flux algebra, Bianchi identities and Freed-Witten anomalies in F-theory compactifications

    Aldazabal, G; Rosabal, J A


    We discuss the structure of 4D gauged supergravity algebras corresponding to globally non-geometric compactifications of F-theory, admitting a local geometric description in terms of (strongly coupled) 10D supergravity. By starting with the well known algebra of gauge generators associated to non-geometric type IIB fluxes, we derive a full algebra containing all, closed RR and NSNS, geometric and non-geometric dual fluxes. We achieve this generalization by a systematic application of SL(2,Z) duality transformations and by taking care of the spinorial structure of the fluxes. The resulting algebra encodes much information about the higher dimensional theory. In particular, tadpole equations and Bianchi identities are obtainable as Jacobi identities of the algebra. When a sector of magnetized (p,q) 7-branes is included, certain closed axions are gauged by the U(1) transformations on the branes. We indicate how the diagonal gauge generators of the branes can be incorporated into the full algebra, and show that F...

  20. Bitopological spaces theory, relations with generalized algebraic structures and applications

    Dvalishvili, Badri


    This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, a

  1. Lie algebraic methods for particle accelerator theory

    The problem of determining charged particle behavior in electromagnetic fields falls within the realm of Hamiltonian dynamics. Consequently, the motion of a charged particle in an accelerator is amenable to description using a variety of the mathematical structures inherent to a Hamiltonian system. Amongst the most useful of these are a hierarchy of Lie algebras and Lie groups defined via the Poisson bracket. In this thesis new applications are made of several concepts from the theory of Lie groups and Lie algebras to certain types of calculations encountered in accelerator science. A variety of techniques are introduced from the theory of Lie algebras which prove useful in developing a description of charged particle motion. Applications of these techniques are then made. A preponderence of this thesis concerns itself with computation of particle trajectories using Lie algebraic methods. An analytical perturbation method for computing particle trajectories is developed and application made to a variety of beam-line elements common in accelerators. In addition, methods for numerical computations based on a Lie algebraic formalism are introduced. An algebraically based tracking code (MARYLIE) is presented as an example of the economy of calculation made possible through use of Lie algebraic methods. This code is designed to perform ray traces through beam lines (comprised of any of a variety of common elements) accurately through nonlinear terms of third order in deviations from beam-line design values. Comparison is made with current matrix theories (which generally include only second order nonlinearities)

  2. Hopf algebras and topological recursion

    We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293–309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347–452). (paper)

  3. Spin structures on algebraic curves and their applications in string theories

    The free fields on a Riemann surface carrying spin structures live on an unramified r-covering of the surface itself. When the surface is represented as an algebraic curve related to the vanishing of the Weierstrass polynomial, its r-coverings are algebraic curves as well. We construct explicitly the Weierstrass polynomial associated to the r-coverings of an algebraic curve. Using standard techniques of algebraic geometry it is then possible to solve the inverse Jacobi problem for the odd spin structures. As an application we derive the partition functions of bosonic string theories in many examples, including two general curves of genus three and four. The partition functions are explicitly expressed in terms of branch points apart from a factor which is essentially a theta constant. 53 refs., 4 figs. (Author)

  4. Threshold complexes and connections to number theory

    Pakianathan, Jonathan; Winfree, Troy


    In this paper we study quota complexes (or equivalently in the case of scalar weights, threshold complexes) and how the topology of these quota complexes changes as the quota is changed. This problem is a simple ``linear\\" version of the general question in Morse Theory of how the topology of a space varies with a parameter. We give examples of natural and basic quota complexes where this problem frames questions about the distribution of primes, squares and divisors in number t...

  5. Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry

    Mulmuley, Ketan D


    This article belongs to a series on geometric complexity theory (GCT), an approach to the P vs. NP and related problems through algebraic geometry and representation theory. The basic principle behind this approach is called the flip. In essence, it reduces the negative hypothesis in complexity theory (the lower bound problems), such as the P vs. NP problem in characteristic zero, to the positive hypothesis in complexity theory (the upper bound problems): specifically, to showing that the problems of deciding nonvanishing of the fundamental structural constants in representation theory and algebraic geometry, such as the well known plethysm constants, belong to the complexity class P. In this article, we suggest a plan for implementing the flip, i.e., for showing that these decision problems belong to P. This is based on the reduction of the preceding complexity-theoretic positive hypotheses to mathematical positivity hypotheses: specifically, to showing that there exist positive formulae--i.e. formulae with ...

  6. Teachers' subjective theories on algebra

    Meinke, Julia


    This project concerns the impact of the belief system of teachers on their algebra teaching practices in secondary education. The first step is to reconstruct this belief system. The methodological framework is provided by the Research Project Subjective Theories (RPST). The research design is described here.

  7. Duality theories for Boolean algebras with operators

    Givant, Steven


    In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as a hybrid of the first two. This text will be of interest to graduate students and researchers in the fields of mathematics, computer science, logic, and philosophy who are interested in exploring special or general classes of Boolean algebras with operators. Readers should be familiar with the basic arithmetic and theory of Boolean algebras, as well as the fundamentals of point-set topology.

  8. The $K$-theory of real graph $C*$-algebras

    Boersema, Jeffrey L.


    In this paper, we will introduce real graph algebras and develop the theory to the point of being able to calculate the $K$-theory of such algebras. The $K$-theory situation is significantly more complicated than in the case for complex graph algebras. To develop the long exact sequence to compute the $K$-theory of a real graph algebra, we need to develop a generalized theory of crossed products for real C*-algebras for groups with involution. We also need to deal with the additional algebrai...

  9. The Adapted Ordering Method for the Representation Theory of Lie Algebras and Superalgebras and their Generalizations

    Gato-Rivera, Beatriz


    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.

  10. Imperfect Cloning Operations in Algebraic Quantum Theory

    Kitajima, Yuichiro


    No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal -imperfect cloning operation which tolerates a finite loss of fidelity in the cloned state, and show that an individual system's algebra of observables is abelian if and only if there is a universal -imperfect cloning operation in the case where the loss of fidelity is less than . Therefore in this case no universal -imperfect cloning operation is possible in algebraic quantum theory.

  11. KK -theory and spectral flow in von Neumann algebras

    Kaad, Jens; Nest, Ryszard; Rennie, Adam


    We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko (J). Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable...

  12. Clifford Algebras in relativistic quantum mechanics and in the gauge theory of electromagnetism

    A Clifford Algebra is an algebra associated with a finite-dimensional vector space and a symmetric form on that space. It contains a multiplicative subgroup, the group of spinors, which is related to the group of orthogonal transformations of the vector space. This group may act on the algebra via multiplication on the left or right, or by the adjoint action. First, the author considers the problem of classifying the orbits of these actions in the algebras C(3,1) and C(3,2). For a ceratin subclass of orbit this problem is completely solved and the isotropy groups for elements in these orbits are determined. After writing the Dirac and Maxwell equations in terms of Clifford Albebras, the author shows how a classification of the solutions to these equations is related to the orbit and isotropy group calculations. Finally, he shows how Clifford algebras may be used to define spinor and r-vector fields on manifolds, gradients of such fields, and other more familiar concepts from differential geometry. The end result is that the calculations for C(3,1) and C(3,2) may be applied to fields on space-time and on the five-dimensional space of the gauge theory of electromagnetism, respectively. This gauge theory also allows us to relate Einstein's equations for free space to Maxwell's equations in a natural manner

  13. Extended Conformal Algebra and Non-commutative Geometry in Particle Theory

    Chagas-Filho, W.


    We show how an off shell invariance of the massless particle action allows the construction of an extension of the conformal space-time algebra and induces a non-commutative space-time geometry in bosonic and supersymmetric particle theories.

  14. Linear Algebra and Smarandache Linear Algebra

    Vasantha, Kandasamy


    The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...

  15. Spectral theory of linear operators and spectral systems in Banach algebras

    Müller, Vladimir


    This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg...

  16. Algebraic conformal quantum field theory in perspective

    Rehren, Karl-Henning


    Conformal quantum field theory is reviewed in the perspective of Axiomatic, notably Algebraic QFT. This theory is particularly developped in two spacetime dimensions, where many rigorous constructions are possible, as well as some complete classifications. The structural insights, analytical methods and constructive tools are expected to be useful also for four-dimensional QFT.

  17. Algebraic perturbation theory for singular potentials

    A purely algebraic theory based on dynamical groups is developed. It allows one to determine the energy shifts without taking any matrix elements. In particular potentials of the form 1/rN and rN are treated explicitly, some examples which cannot be calculated by the usual perturbation theory are discussed. ((orig.))

  18. Algebra


    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.

  19. The algebraic crossing number and the braid index of knots and links

    Kawamuro, Keiko


    It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable of K and for the connect sum of K and L.

  20. Elliptic Tales Curves, Counting, and Number Theory

    Ash, Avner


    Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of 1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem. The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from

  1. Properties of Quaternion Algebra over the Real Number Field and Zp

    QIN Ying-bing


    The ring of quaternion over R, denoted by R[i,j,k], is a quaternion algebra. In this paper, the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed, and the order of its unit group is determined. Lastly, another ring isomorphism of M2(Zp) and the quaternion algebra over Zp when p satisfies some particular conditions are presented.

  2. Conference on Number Theory and Arithmetic Geometry

    Silverman, Joseph; Stevens, Glenn; Modular forms and Fermat’s last theorem


    This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, ...

  3. Understanding geometric algebra for electromagnetic theory

    Arthur, John W


    "This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison"--Provided by publisher.

  4. Virtual Betti numbers of real algebraic varieties

    McCrory, Clint; Parusinski, Adam


    The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\\Y) + beta_i(Y).

  5. Ideal Theory in BCK/BCI-Algebras Based on Soft Sets and -Structures

    Young Bae Jun; Min Su Kang; Kyoung Ja Lee


    Based on soft sets and -structures, the notion of (closed) -ideal over a BCI-algebra is introduced, and related properties are investigated. Relations between -BCI-algebras and -ideals are established. Characterizations of a (closed) -ideal over a BCI-algebra are provided. Conditions for an -ideal to be an -BCI-algebra are considered.

  6. Categorical Algebra and its Applications


    Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.

  7. Supertropical algebra

    Izhakian, Zur; Rowen, Louis


    We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...

  8. The number of simple modules of a cellular algebra

    LI Weixia; XI Changchang


    Let n be a natural number, and let A be an indecomposable cellular algebra such that the spectrum of its Cartan matrix C is of theform {n, 1,..., 1}. In general, not every natural number could be the number of non-isomorphic simple modules over such a cellular algebra. Thus, two natural questions arise: (1) which numbers could be the number of non-isomorphic simple modules over such a cellular algebra A ? (2) Given such a number, is there a cellular algebra such that its Cartan matrix has the desired property ? In this paper, we shall completely answer the first question, and give a partial answer to the second question by constructing cellular algebras with the pre-described Cartan matrix.

  9. Quantum field theory on toroidal topology: algebraic structure and applications

    Khanna, F C; Malbouisson, J M C; Santana, A E


    The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordstr\\"om, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particles physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matted physics.

  10. Computing with impure numbers - Automatic consistency checking and units conversion using computer algebra

    Stoutemyer, D. R.


    The computer algebra language MACSYMA enables the programmer to include symbolic physical units in computer calculations, and features automatic detection of dimensionally-inhomogeneous formulas and conversion of inconsistent units in a dimensionally homogeneous formula. Some examples illustrate these features.

  11. Classical and quantum Kummer shape algebras

    Odzijewicz, A.; Wawreniuk, E.


    We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.

  12. Representation theory a homological algebra point of view

    Zimmermann, Alexander


      Introducing the representation theory of groups and finite dimensional algebras, this book first studies basic non-commutative ring theory, covering the necessary background of elementary homological algebra and representations of groups to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings ...

  13. Number Theory : A Seminar held at the Graduate School and University Center of the City University of New York

    Chudnovsky, Gregory; Cohn, Harvey; Nathanson, Melvyn


    The New York Number Theory Seminar was organized in 1982 to provide a forum for the presentation and discussion of recent advances in higher arithmetic and its applications. Papers included in this volume are based on the lectures presented by their authors at the Seminar at the Graduate Center of C.U.N.Y. in 1985-88. Papers in the volume cover a wide spectrum of number theoretic topics ranging from additive number theory and diophantine approximations to algebraic number theory and relations with algebraic geometry and topology.

  14. Number theory and the periodicity of matter

    Boeyens, Jan CA


    The main purpose of the book is to communicate a fundamental principle to the scientific world. The eventual impact of the subject matter is considered to be much wider than the readership of the preliminary accounts which have been published. The number principle at issue is known to be of wide general interest and the book has also been written to be accessible to nonspecialists. The potential readership should extend way beyond academic scientists. The discovery described in this book could be of seminal significance, also in other fields where the golden ratio is known to be of fundamental importance. The most obvious connection is with Fibonacci phylotaxis in the study of botanical growth and the number basis of DNA coding. In another context it may impinge on crystallographic periodicity and the structure of quasicrystals. These topics are beyond the scope of this book and hence it is all the more important that the power of number theory to describe physical systems be disseminated more widely.

  15. Quantum exchange algebra and exact operator solution of A sub 2 -Toda field theory

    Takimoto, Y; Kurokawa, H; Fujiwara, T


    Locality is analyzed for Toda field theories by noting novel chiral description in the conventional non-chiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to non-trivial relations among the R-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for a A sub 2 -system by extending the algebraic method developed for the Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.

  16. Quantum Exchange Algebra and Exact Operator Solution of $A_{2}$-Toda Field Theory

    Takimoto, Y; Kurokawa, H; Fujiwara, T


    Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to nontrivial relations among the ${\\cal R}$-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for $A_2$-system by extending the algebraic method developed for Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.

  17. Quantum double actions on operator algebras and orbifold quantum field theories

    Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1 dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitary locally compact groups and our methods are adapted to chiral theories on the circle. (orig.)

  18. Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics

    Ismail, Mourad


    These are the proceedings of the conference "Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics" held at the Department of Mathematics, University of Florida, Gainesville, from November 11 to 13, 1999. The main emphasis of the conference was Com­ puter Algebra (i. e. symbolic computation) and how it related to the fields of Number Theory, Special Functions, Physics and Combinatorics. A subject that is common to all of these fields is q-series. We brought together those who do symbolic computation with q-series and those who need q-series in­ cluding workers in Physics and Combinatorics. The goal of the conference was to inform mathematicians and physicists who use q-series of the latest developments in the field of q-series and especially how symbolic computa­ tion has aided these developments. Over 60 people were invited to participate in the conference. We ended up having 45 participants at the conference, including six one hour plenary speakers and 28 half hour speakers. T...

  19. Lie algebras and applications

    Iachello, Francesco


    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...

  20. Clifford algebra in finite quantum field theories

    We consider the most general power counting renormalizable and gauge invariant Lagrangean density L invariant with respect to some non-Abelian, compact, and semisimple gauge group G. The particle content of this quantum field theory consists of gauge vector bosons, real scalar bosons, fermions, and ghost fields. We assume that the ultimate grand unified theory needs no cutoff. This yields so-called finiteness conditions, resulting from the demand for finite physical quantities calculated by the bare Lagrangean. In lower loop order, necessary conditions for finiteness are thus vanishing beta functions for dimensionless couplings. The complexity of the finiteness conditions for a general quantum field theory makes the discussion of non-supersymmetric theories rather cumbersome. Recently, the F = 1 class of finite quantum field theories has been proposed embracing all supersymmetric theories. A special type of F = 1 theories proposed turns out to have Yukawa couplings which are equivalent to generators of a Clifford algebra representation. These algebraic structures are remarkable all the more than in the context of a well-known conjecture which states that finiteness is maybe related to global symmetries (such as supersymmetry) of the Lagrangean density. We can prove that supersymmetric theories can never be of this Clifford-type. It turns out that these Clifford algebra representations found recently are a consequence of certain invariances of the finiteness conditions resulting from a vanishing of the renormalization group β-function for the Yukawa couplings. We are able to exclude almost all such Clifford-like theories. (author)

  1. Quantum field theories on algebraic curves. I. Additive bosons

    Using Serre's adelic interpretation of cohomology, we develop a 'differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.

  2. Quantum field theories on algebraic curves. I. Additive bosons

    Takhtajan, Leon A.


    Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.

  3. Finite automata, their algebras and grammars towards a theory of formal expressions

    Büchi, J Richard


    The author, who died in 1984, is well-known both as a person and through his research in mathematical logic and theoretical computer science. In the first part of the book he presents the new classical theory of finite automata as unary algebras which he himself invented about 30 years ago. Many results, like his work on structure lattices or his characterization of regular sets by generalized regular rules, are unknown to a wider audience. In the second part of the book he extends the theory to general (non-unary, many-sorted) algebras, term rewriting systems, tree automata, and pushdown automata. Essentially Büchi worked independent of other rersearch, following a novel and stimulating approach. He aimed for a mathematical theory of terms, but could not finish the book. Many of the results are known by now, but to work further along this line presents a challenging research program on the borderline between universal algebra, term rewriting systems, and automata theory. For the whole book and aga...

  4. Introduction to number theory

    Vazzana, Anthony; Garth, David


    One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topics.

  5. Algebraic Independence of Certain Generalized Mahler Type Numbers

    Yao Chen ZHU


    In this paper the generalized Mahler type number Mh(g;A,T) is defined, and in the case of multiplicatively dependent parameters gi, hi(1≤ I≤ s) the algebraic independence of the numbers Mh, (gi; A, T)(1≤I≤s) is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.

  6. The algebra of space-time as basis of a quantum field theory of all fermions and interactions

    In this thesis a construction of a grand unified theory on the base of algebras of vector fields on a Riemannian space-time is described. Hereby from the vector and covector fields a Clifford-geometrical algebra is generated. (HSI)

  7. An experimental investigation of the normality of irrational algebraic numbers

    Nielsen, Johan Sejr Brinch; Simonsen, Jakob Grue


    We investigate the distribution of digits of large prefixes of the expansion of irrational algebraic numbers to different bases. We compute 2.318 bits of the binary expansions (corresponding to 2.33.108 decimals) of the 39 least Pisot-Vijayaraghavan numbers, the 47 least known Salem numbers, the...... blocks for each number to bases 2, 3, 5, 7 and 10, as well as the maximum relative frequency deviation from perfect equidistribution. We use the two statistics to perform tests at significance level α = 0.05, respectively, maximum deviation threshold α = 0.05. Our results suggest that if Borel......'s conjecture-that all irrational algebraic numbers are normal-is true, then it may have an empirical base: The distribution of digits in algebraic numbers appears close to equidistribution for large prefixes of their expansion. Of the 121 algebraic numbers studied, all numbers passed the maximum relative...

  8. Methods of algebraic geometry in control theory

    Falb, Peter


    "Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is qui...

  9. The Green formula and heredity of algebras


    [1]Green, J. A., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 1995, 120: 361-377.[2]Ringel, C. M., Green's theorem on Hall algebras, in Representations of Algebras and Related Topics, CMS Conference Proceedings 19, Providence, 1996, 185-245.[3]Xiao J., Drinfeld double and Ringel-Green theory of Hall Algebras, J. Algebra, 1997, 190: 100-144.[4]Sevenhant, B., Van den Bergh, M., A relation between a conjecture of Kac and the structure of the Hall algebra,J. Pure Appl. Algebra, 2001, 160: 319-332.[5]Deng B., Xiao, J., On double Ringel-Hall algebras, J. Algebra, 2002, 251: 110-149.

  10. On the Algebraic K Theory of the Massive D8 and M9-Branes

    Vancea, Ion V.

    In this paper we review some basic relations of algebraic K theory and we formulate them in the language of D-branes. Then we study the relation between the D8-branes wrapped on an orientable compact manifold W in a massive Type IIA supergravity background and the M9-branes wrapped on a compact manifold Z in a massive d=11 supergravity background from the K-theoretic point of view. By interpreting the D8-brane charges as elements of K0(C(W)) and the (inequivalent classes of) spaces of gauge fields on the M9-branes as the elements of K0(C(Z)x{¯ {k}*}G) where G is a one-dimensional compact group, a connection between charges and gauge fields is argued to exists. This connection could be realized as a composition map between the corresponding algebraic K theory groups.

  11. Lower bounds for algebraic connectivity of graphs in terms of matching number or edge covering number

    Xu, Jing; Fan, Yi-Zheng; Tan, Ying-Ying


    In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number.

  12. Reduced Chern-Simons Quiver Theories and Cohomological 3-Algebra Models

    DeBellis, Joshua


    We study the BPS spectrum and vacuum moduli spaces in dimensional reductions of Chern-Simons-matter theories with N>=2 supersymmetry to zero dimensions. Our main example is a matrix model version of the ABJM theory which we relate explicitly to certain reduced 3-algebra models. We find the explicit maps from Chern-Simons quiver matrix models to dual IKKT matrix models. We address the problem of topologically twisting the ABJM matrix model, and along the way construct a new twist of the IKKT model. We construct a cohomological matrix model whose partition function localizes onto a moduli space specified by 3-algebra relations which live in the double of the conifold quiver. It computes an equivariant index enumerating framed BPS states with specified R-charges which can be expressed as a combinatorial sum over certain filtered pyramid partitions.

  13. Field algebra, Hilbert space and observables in two-dimensional higher-derivative field theories

    We discuss structural aspects related to the field algebra of two-dimensional higher-derivative quantum field theories. We present general selection criteria for the proper field subalgebra that generates Wightman functions satisfying the asymptotic factorization property and which define a semi-definite inner product Hilbert space. The positive definite inner product Hilbert space, which contains as a subspace of states the general Wightman functions of the corresponding standard canonical models, is a quotient space obtained by equivalence classes. For higher-derivative local gauge theories, besides the Lowenstein-Swieca condition, an additional condition must be imposed on the field algebra in order to obtain a physical subspace of states satisfying a reasonable set of physically meaningful axioms. (author)

  14. Analytic number theory

    Matsumoto, Kohji


    The book includes several survey articles on prime numbers, divisor problems, and Diophantine equations, as well as research papers on various aspects of analytic number theory such as additive problems, Diophantine approximations and the theory of zeta and L-function Audience Researchers and graduate students interested in recent development of number theory

  15. A note on Quarks and numbers theory

    Hage-Hassan, Mehdi


    We express the basis vectors of Cartan fundamental representations of unitary groups by binary numbers. We determine the expression of Gel'fand basis of SU (3) based on the usual subatomic quarks notations and we represent it by binary numbers. By analogy with the mesons and quarks we find a new property of prime numbers.

  16. Operator Algebras and Conformal Field Theory III. Fusion of positive energy representations of LSU(N) using bounded operators

    Wassermann, Antony


    Fusion of positive energy representations is defined using Connes' tensor product for bimodules over a von Neumann algebra. Fusion is computed using the analytic theory of primary fields and explicit solutions of the Knizhnik-Zamolodchikov equation.

  17. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

    Ion C. Baianu


    Full Text Available A novel algebraic topology approach to supersymmetry (SUSY and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin models with the extended quantum symmetry of entangled, 'string-net condensed' (ground states.

  18. Wild Pfister forms over Henselian fields, K-theory, and conic division algebras

    Garibaldi, Skip


    The epicenter of this paper concerns Pfister quadratic forms over a field $F$ with a Henselian discrete valuation. All characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but $F$ does not. We also prove results about round quadratic forms, composition algebras, generalizations of composition algebras we call conic algebras, and central simple associative symbol algebras. Finally we give relationships between these objects and Kato's filtration on the Milnor $K$-groups of $F$.

  19. Matrices and linear algebra

    Schneider, Hans


    Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t

  20. Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model

    Zahabi, Ali


    We review the algebraic and analytic aspects of the conformal field theory (CFT) and its relation to the stochastic Loewner evolution (SLE) in an example of the Ising model. We obtain the scaling limit of the correlation functions of Ising free fermions on an arbitrary simply connected two-dimensional domain $D$. Then, we study the analytic and algebraic aspects of the fermionic CFT on $D$, using the Fock space formalism of fields, and the Clifford vertex operator algebra (VOA). These constructions lead to the conformal field theory of the Fock space fields and the fermionic Fock space of states and their relations in case of the Ising free fermions. Furthermore, we investigate the conformal structure of the fermionic Fock space fields and the Clifford VOA, namely the operator product expansions, correlation functions and differential equations. Finally, by using the Clifford VOA and the fermionic CFT, we investigate a rigorous realization of the CFT/SLE correspondence in the Ising model. First, by studying t...

  1. Introduction to the theory of abstract algebras

    Pierce, Richard S


    Intended for beginning graduate-level courses, this text introduces various aspects of the theory of abstract algebra. The book is also suitable as independent reading for interested students at that level as well as a primary source for a one-semester course that an instructor may supplement to expand to a full year. Author Richard S. Pierce, a Professor of Mathematics at Seattle's University of Washington, places considerable emphasis on applications of the theory and focuses particularly on lattice theory.After a preliminary review of set theory, the treatment presents the basic definitions

  2. Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation

    Rim, Chaiho


    AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with Matsuo's interwiner we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory which involves summation of functions over Young diagrams.

  3. Algebraic combinatorics and coinvariant spaces

    Bergeron, Francois


    Written for graduate students in mathematics or non-specialist mathematicians who wish to learn the basics about some of the most important current research in the field, this book provides an intensive, yet accessible, introduction to the subject of algebraic combinatorics. After recalling basic notions of combinatorics, representation theory, and some commutative algebra, the main material provides links between the study of coinvariant-or diagonally coinvariant-spaces and the study of Macdonald polynomials and related operators. This gives rise to a large number of combinatorial questions r

  4. Representations of Conformal Nets, Universal C*-Algebras and K-Theory

    Carpi, Sebastiano; Conti, Roberto; Hillier, Robin; Weiner, Mihály


    We study the representation theory of a conformal net {{A}} on S 1 from a K-theoretical point of view using its universal C*-algebra {C^*({A})}. We prove that if {{A}} satisfies the split property then, for every representation π of {{A}} with finite statistical dimension, {π(C^*({A}))} is weakly closed and hence a finite direct sum of type I∞ factors. We define the more manageable locally normal universal C*-algebra {C_ln^*({A})} as the quotient of {C^*({A})} by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if {{A}} is completely rational with n sectors, then {C_ln^*({A})} is a direct sum of n type I∞ factors. Its ideal {{K}_{A}} of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of {C^*({A})} with finite statistical dimension act on {{K}_{A}}, giving rise to an action of the fusion semiring of DHR sectors on {K_0({K}_{A})}. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.

  5. On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces

    In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker-Campbell-Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X1, X2, ..., Xn] with the poly-vector-valued coordinates X123...n in noncommutative Clifford spaces given by [X1, X2, ..., Xn] = n!X123...n. The large N limit of n-ary commutators of n hyper-matrices leads to Eguchi-Schild p-brane actions for p + 1 = n. A noncomutative n-ary . product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu-Poisson brackets.

  6. On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces

    Castro, Carlos, E-mail: [Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314 (United States)


    In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker-Campbell-Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X{sup 1}, X{sup 2}, ..., X{sup n}] with the poly-vector-valued coordinates X{sup 123...n} in noncommutative Clifford spaces given by [X{sup 1}, X{sup 2}, ..., X{sup n}] = n!X{sup 123...n}. The large N limit of n-ary commutators of n hyper-matrices X{sub i{sub 1i{sub 2...i{sub n}}}} leads to Eguchi-Schild p-brane actions for p + 1 = n. A noncomutative n-ary . product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu-Poisson brackets.

  7. Clifford Algebras and magnetic monopoles

    It is known that the introduction of magnetic monopolies in electromagnetism does still present formal problems from the point of view of classical field theory. The author attempts to overcome at least some of them by making recourse to the Clifford Algebra formalism. In fact, while the events of a two-dimensional Minkowski space-time M(1,1) are sufficiently well represented by ordinary Complex Numbers, when dealing with the events of the four-dimensional Minkowski space M(1,3)identical to M/sub 4/ one has of course to look for hypercomplex numbers or, more generally, for the elements of a Clifford Algebra. The author uses the Clifford Algebras in terms of ''multivectors'', and in particular by Hestenes' language, which suits space-time quite well. He recalls that the Clifford product chiγ is the sum of the internal product chi . γ and of the wedge product chiΛγ


    Labourie, François


    We define a Poisson Algebra called the {\\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\\mathsf{SL}_n(\\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symple...

  9. Applied number theory

    Niederreiter, Harald


    This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas.  Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc.  Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters...

  10. On the algebra of deformed differential operators, and induced integrable Toda field theory

    We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalised KdV hierarchy. We focus in particular the first leading orders of this q-deformed hierarchy namely the q-KdV and q-Boussinesq integrable systems. We also present the q-generalisation of the conformal transformations of the currents un, n ≥ 2 and discuss the primary condition of the fields wn, n ≥ 2 by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented. (author)