Download or read book Lie Algebras with Triangular Decompositions written by Robert V. Moody and published by Wiley-Interscience. This book was released on 1995-04-17 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt: Imparts a self-contained development of the algebraic theory of Kac-Moody algebras, their representations and close relatives--the Virasoro and Heisenberg algebras. Focuses on developing the theory of triangular decompositions and part of the Kac-Moody theory not specific to the affine case. Also covers lattices, and finite root systems, infinite-dimensional theory, Weyl groups and conjugacy theorems.

Download or read book Quantum Mechanics via Lie Algebras written by Arnold Neumaier and published by Walter de Gruyter GmbH & Co KG. This book was released on 2024-10-07 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph introduces mathematicians, physicists, and engineers to the ideas relating quantum mechanics and symmetries - both described in terms of Lie algebras and Lie groups. The exposition of quantum mechanics from this point of view reveals that classical mechanics and quantum mechanics are very much alike. Written by a mathematician and a physicist, this book is (like a math book) about precise concepts and exact results in classical mechanics and quantum mechanics, but motivated and discussed (like a physics book) in terms of their physical meaning. The reader can focus on the simplicity and beauty of theoretical physics, without getting lost in a jungle of techniques for estimating or calculating quantities of interest.

Download or read book Highlights in Lie Algebraic Methods written by Anthony Joseph and published by Springer Science & Business Media. This book was released on 2011-10-20 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of expository and research articles that highlight the various Lie algebraic methods used in mathematical research today. Key topics discussed include spherical varieties, Littelmann Paths and Kac–Moody Lie algebras, modular representations, primitive ideals, representation theory of Artin algebras and quivers, Kac–Moody superalgebras, categories of Harish–Chandra modules, cohomological methods, and cluster algebras.

Download or read book Lie Superalgebras and Enveloping Algebras written by Ian Malcolm Musson and published by American Mathematical Soc.. This book was released on 2012-04-04 with total page 512 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations. The book begins with five chapters on the basic properties of Lie superalgebras, including explicit constructions for all the classical simple Lie superalgebras. Borel subalgebras, which are more subtle in this setting, are studied and described. Contragredient Lie superalgebras are introduced, allowing a unified approach to several results, in particular to the existence of an invariant bilinear form on $\mathfrak{g}$. The enveloping algebra of a finite dimensional Lie superalgebra is studied as an extension of the enveloping algebra of the even part of the superalgebra. By developing general methods for studying such extensions, important information on the algebraic structure is obtained, particularly with regard to primitive ideals. Fundamental results, such as the Poincare-Birkhoff-Witt Theorem, are established. Representations of Lie superalgebras provide valuable tools for understanding the algebras themselves, as well as being of primary interest in applications to other fields. Two important classes of representations are the Verma modules and the finite dimensional representations. The fundamental results here include the Jantzen filtration, the Harish-Chandra homomorphism, the Sapovalov determinant, supersymmetric polynomials, and Schur-Weyl duality. Using these tools, the center can be explicitly described in the general linear and orthosymplectic cases. In an effort to make the presentation as self-contained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.

Download or read book Lie Groups and Lie Algebras III written by A.L. Onishchik and published by Springer Science & Business Media. This book was released on 1994-07-12 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive and modern account of the structure and classification of Lie groups and finite-dimensional Lie algebras, by internationally known specialists in the field. This Encyclopaedia volume will be immensely useful to graduate students in differential geometry, algebra and theoretical physics.

Download or read book Projective Modules over Lie Algebras of Cartan Type written by Daniel Ken Nakano and published by American Mathematical Soc.. This book was released on 1992 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper investigates the question of linkage and block theory for Lie algebras of Cartan type. The second part of the paper deals mainly with block structure and projective modules of Lies algebras of types W and K.

Download or read book Introduction to Lie Algebras written by K. Erdmann and published by Springer Science & Business Media. This book was released on 2006-09-28 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

Download or read book Yangians and Classical Lie Algebras written by Alexander Molev and published by American Mathematical Soc.. This book was released on 2007 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St. Petersburg's school of mathematical physics in the 1980s. This book is an introduction to the theory of Yangians and twisted Yangians, with a particular emphasis on the relationship with the classical matrix Lie algebras.

Download or read book Representations of Semisimple Lie Algebras in the BGG Category O written by James E. Humphreys and published by American Mathematical Soc.. This book was released on 2021-07-14 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra g g over C C. The setting is the module category O O introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for g g such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of g g. Basic techniques in category O O such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: D D-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category O O, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.

Download or read book Identical Relations in Lie Algebras written by Yuri Bahturin and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-08-23 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: This updated edition of a classic title studies identical relations in Lie algebras and also in other classes of algebras, a theory with over 40 years of development in which new methods and connections with other areas of mathematics have arisen. New topics covered include graded identities, identities of algebras with actions and coactions of various Hopf algebras, and the representation theory of the symmetric and general linear group.

Download or read book Lie Groups and Algebraic Groups written by Arkadij L. Onishchik and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in 1967/68. Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Our main sources were A. Borel's paper [34], C. ChevalIey's seminar [14], seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N. Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we have completely rearranged these notes and added two new chapters: "Lie groups" and "Real semisimple Lie groups". Several traditional topics of Lie algebra theory, however, are left entirely disregarded, e.g. universal enveloping algebras, characters of linear representations and (co)homology of Lie algebras. A distinctive feature of this book is that almost all the material is presented as a sequence of problems, as it had been in the first draft of the seminar's notes. We believe that solving these problems may help the reader to feel the seminar's atmosphere and master the theory. Nevertheless, all the non-trivial ideas, and sometimes solutions, are contained in hints given at the end of each section. The proofs of certain theorems, which we consider more difficult, are given directly in the main text. The book also contains exercises, the majority of which are an essential complement to the main contents.

Download or read book Lie Theory and Its Applications in Physics written by Vladimir Dobrev and published by Springer Nature. This book was released on 2023-01-29 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents modern trends in the area of symmetries and their applications based on contributions to the Workshop "Lie Theory and Its Applications in Physics" held in Sofia, Bulgaria (on-line) in June 2021. Traditionally, Lie theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrization and symmetries are meant in their widest sense, i.e., representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators, special functions, and others. Furthermore, the necessary tools from functional analysis are included. This is a big interdisciplinary and interrelated field. The topics covered in this Volume are the most modern trends in the field of the Workshop: Representation Theory, Symmetries in String Theories, Symmetries in Gravity Theories, Supergravity, Conformal Field Theory, Integrable Systems, Quantum Computing and Deep Learning, Entanglement, Applications to Quantum Theory, Exceptional quantum algebra for the standard model of particle physics, Gauge Theories and Applications, Structures on Lie Groups and Lie Algebras. This book is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists, including researchers and graduate students interested in Lie Theory.

Download or read book Recent Advances in Operator Theory Operator Algebras and their Applications written by Dumitru Gaspar and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers peer-reviewed articles from the 19th International Conference on Operator Theory, Summer 2002. It contains recent developments in a broad range of topics from operator theory, operator algebras and their applications, particularly to differential analysis, complex functions, ergodic theory, mathematical physics, matrix analysis, and systems theory. The book covers a large variety of topics including single operator theory, C*-algebras, diffrential operators, integral transforms, stochastic processes and operators, and more.

Download or read book Algebraic Structures and Moduli Spaces written by Jacques Hurtubise and Eyal Markman and published by American Mathematical Soc.. This book was released on with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains recent and exciting developments on the structure of moduli spaces, with an emphasis on the algebraic structures that underlie this structure. Topics covered include Hilbert schemes of points, moduli of instantons, coherent sheaves and their derived categories, moduli of flat connections, Hodge structures, and the topology of affine varieties. Two beautiful series of lectures are a particularly fine feature of the book. One is an introductory series by Manfred Lehn on the topology and geometry of Hilbert schemes of points on surfaces, and the other, by Hiraku Nakajima and Kota Yoshioka, explains their recent work on the moduli space of instantons over ${\mathbb R 4$. The material is suitable for graduate students and researchers interested in moduli spaces in algebraic geometry, topology, and mathematical physics.

Download or read book Clifford Algebras and their Applications in Mathematical Physics written by Rafał Abłamowicz and published by Springer Science & Business Media. This book was released on 2000 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras. algebras and their applications in physics. Algebraic geometry, cohomology, non-communicative spaces, q-deformations and the related quantum groups, and projective geometry provide the basis for algebraic topics covered. Physical applications and extensions of physical theories such as the theory of quaternionic spin, a projective theory of hadron transformation laws, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems. Treatment of the structure theory of quantum Clifford algebras, the connection to logic, group representations, and computational techniques including symbolic calculations and theorem proving rounds out the presentation.

Download or read book Clifford Algebras and Lie Theory written by Eckhard Meinrenken and published by Springer Science & Business Media. This book was released on 2013-02-28 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.

Download or read book Algebras Rings and Their Representations written by Alberto Facchini and published by World Scientific. This book was released on 2006 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: Surveying the most influential developments in the field, this proceedings reviews the latest research on algebras and their representations, commutative and non-commutative rings, modules, conformal algebras, and torsion theories. The volume collects stimulating discussions from world-renowned names including Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith. Sample Chapter(s). Chapter 1: Some Coreflective Categories of Topological Modules (221 KB). Contents: Krull Monoids and Their Application in Module Theory (A Facchini); Infinite Progenerator Sums (A Facchini & L S Levy); Quadratic Algebras of Skew Type (E Jespers & J Okn nski); Representation Type of Commutative Noetherian Rings (Introduction) (L Klingler & L S Levy); Corner Ring Theory: A Generalization of Peirce Decompositions (T-Y Lam); Quasideterminants and Right Roots of Polynomials Over Division Rings (B L Osofsky); Injective Dimension Relative to a Torsion Theory (P F Smith); and other papers. Readership: Algebraists, mathematicians interested in the connections between algebra and other fields, and graduate students interested in algebra."