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Author : Peter Schauenburg
Publisher : Reinhard Fischer
Page : 90 pages
File Size : 47,7 Mb
Release : 1992
Category : Mathematics
ISBN : UVA:X002152806
Author : Florin Felix Nichita
Publisher : MDPI
Page : 239 pages
File Size : 40,5 Mb
Release : 2019-01-31
Category : Mathematics
ISBN : 9783038973249
This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms
Author : L.A. Lambe,D.E. Radford
Publisher : Springer Science & Business Media
Page : 314 pages
File Size : 42,7 Mb
Release : 2013-11-22
Category : Mathematics
ISBN : 9781461541097
Chapter 1 The algebraic prerequisites for the book are covered here and in the appendix. This chapter should be used as reference material and should be consulted as needed. A systematic treatment of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of these objects to the extent needed for the book is given. The material here not specifically cited can be found for the most part in [Sweedler, 1969] in one form or another, with a few exceptions. A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book. We have found pointed bialgebras useful in connection with solving the quantum Yang-Baxter equation. For this reason we develop their theory in some detail. The class of examples described in Chapter 6 in connection with the quantum double consists of pointed Hopf algebras. We note the quantized enveloping algebras described Hopf algebras. Thus for many reasons pointed bialgebras are elsewhere are pointed of fundamental interest in the study of the quantum Yang-Baxter equation and objects quantum groups.
Author : Andrew Pressley
Publisher : Cambridge University Press
Page : 246 pages
File Size : 46,5 Mb
Release : 2002-01-17
Category : Mathematics
ISBN : 113943702X
This book comprises an overview of the material presented at the 1999 Durham Symposium on Quantum Groups and includes contributions from many of the world's leading figures in this area. It will be of interest to researchers and will also be useful as a reference text for graduate courses.
Author : Pavel Etingof,Frederic Latour
Publisher : OUP Oxford
Page : 152 pages
File Size : 51,6 Mb
Release : 2005-03-24
Category : Mathematics
ISBN : 9780191523922
The text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups. The book, which contains many detailed proofs and explicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semisimple Lie algebras.
Author : Christian Kassel
Publisher : Springer
Page : 560 pages
File Size : 45,6 Mb
Release : 1995
Category : Mathematics
ISBN : UOM:49015002261700
This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on Drinfeld's recent fundamental contributions. The first part presents in detail the quantum groups attached to SL[subscript 2] as well as the basic concepts of the theory of Hopf algebras. Part Two focuses on Hopf algebras that produce solutions of the Yang-Baxter equation, and on Drinfeld's quantum double construction. In the following part we construct isotopy invariants of knots and links in the three-dimensional Euclidean space, using the language of tensor categories. The last part is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations, culminating in the construction of Kontsevich's universal knot invariant.
Author : Zhongqi Ma
Publisher : World Scientific
Page : 336 pages
File Size : 48,6 Mb
Release : 1993
Category : Science
ISBN : 9810213832
This is the first-ever textbook on the Yang-Baxter equation. A key nonlinear equation for solving two important models in many-body statistical theory - the many-body problem in one dimension with repulsive delta-function interaction presented by Professor Baxter in 1972 - it has become one of the main concerns of physicists and mathematicians in the last ten years. A textbook on this subject which also serves as a reference book is vital for an equation which plays important roles in diverse areas of physics and mathematics like the completely integrable statistical models, conformal field theories, topological field theories, the theory of braid groups, the theory of knots and links, etc. This book arose from lectures given by the author in an attempt to reformulate the results of the rapidly developing research and make the material more accessible. It explains the presentation of the Yang-Baxter equation from statistical models, and expound systematically the meaning and methods of solving for this equation. From the viewpoint of theoretical physics it aims to develop an intuitive understanding of the fundamental knowledge of the Hopf algebras, quantization of Lie bialgebras, and the quantum enveloping algebras, and places emphasis on the introduction of the calculation skill in terms of the physical language.
Author : David E Radford
Publisher : World Scientific
Page : 588 pages
File Size : 40,6 Mb
Release : 2011-12-28
Category : Mathematics
ISBN : 9789814405102
The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph. Contents:PreliminariesCoalgebrasRepresentations of CoalgebrasThe Coradical Filtration and Related StructuresBialgebrasThe Convolution AlgebraHopf AlgebrasHopf Modules and Co-Hopf ModulesHopf Algebras as Modules Over Their Hopf SubalgebrasIntegralsActions by Bialgebras and Hopf AlgebrasQuasitriangular Bialgebras and Hopf AlgebrasThe Drinfel'd Double of a Finite-Dimensional Hopf AlgebraCo-Quasitriangular Bialgebras and Hopf AlgebrasPointed Hopf AlgebrasFinite-Dimensional Hopf Algebras in Characteristic 0 Readership: Undergraduates and researchers in algebra and number theory. Keywords:Hopf Algebras;Coalgebras;Quantum GroupsKey Features:Provides a good foundation for those who wish to study Hopf algebras on their ownProvides a firm foundation for those who are more interested in applications to other areasGives many exercises which suggest connections to exploreReviews: "With this monograph, one of the pioneers of the subject provides a comprehensive introduction to the theory of Hopf algebras. As this theory has made great strides in recent years, such a monograph constitutes a very valuable addition to the literature, especially as there are so far comparatively few textbooks on this topic. Radford's book contains at the end of each chapter a very useful set of chapter notes that discuss these references and therefore provide an entry point to the recent research literature, especially to the extensive literature on the classification of finite–dimensional pointed Hopf algebras, a topic not discussed in any of the other books. For all these reasons, Radford's book is a very valuable new textbook on Hopf algebras that will be frequently used both by students and by researchers." Mathematical Reviews "A big number of exercises of different level of difficulty are proposed along the text, which include in particular special features or applications to a variety of concrete examples, further results and categorical aspects of the corresponding material. Interesting and up-to-date historical and bibliographical comments are provided at the end of each of the sixteen chapters." Zentralblatt MATH
Author : Nicolás Andruskiewitsch,Walter Ricardo Ferrer Santos,Hans-Jürgen Schneider
Publisher : American Mathematical Soc.
Page : 376 pages
File Size : 44,5 Mb
Release : 2000
Category : Hopf algebras
ISBN : 9780821821268
This volume presents the proceedings from the Colloquium on Quantum Groups and Hopf Algebras held in Cordoba (Argentina) in 1999. The meeting brought together researchers who discussed recent developments in Hopf algebras, one of the most important being the influence of quantum groups. Articles offer introductory expositions and surveys on topics of current interest that, to date, have not been available in the current literature. Surveys are included on characteristics of Hopf algebras and their generalizations, biFrobenius algebras, braided Hopf algebras, inner actions and Galois theory, face algebras, and infinitesimal Hopf algebras. The following topics are also covered: existence of integrals, classification of semisimple and pointed Hopf algebras, *-Hopf algebras, dendriform algebras, etc. Non-classical topics are also included, reflecting its applications both inside and outside the theory.
Author : Petr P. Kulish
Publisher : Unknown
Page : 432 pages
File Size : 41,6 Mb
Release : 1992
Category : Mathematics
ISBN : UCSD:31822015075856
The theory of Quantum Groups is a rapidly developing area with numerous applications in mathematics and theoretical physics, e.g. in link and knot invariants in topology, q-special functions, conformal field theory, quantum integrable models. The aim of the Euler Institute's workshops was to review and compile the progress achieved in the different subfields. Near 100 participants came from 14 countries. More than 20 contributions written up for this book contain new, unpublished material and half of them include a survey of recent results in the field (deformation theory, graded differential algebras, contraction technique, knot invariants, q-special functions). FROM THE CONTENTS: V.G. Drinfeld: On Some Unsolved Problems in Quantum Group Theory.- M. Gerstenhaber, A. Giaquinto, S.D. Schack: Quantum Symmetry.- L.I. Korogodsky, L.L. Vaksman: Quantum G-Spaces and Heisenberg Algebra.-J. Stasheff: Differential Graded Lie Algebras, Quasi-Hopf Algebras and Higher Homotopy Algebras.- A. Yu. Alekseev, L.D. Faddeev, M.A. Semenov-Tian-Shansky: Hidden Quantum Groups inside Kac-Moody Algebras.- J.-L. Gervais: Quantum Group Symmetry of 2D Gravity.- T. Kohno: Invariants of 3-Manifolds Based on Conformal Field Theory and Heegaard Splitting.- O. Viro: Moves of Triangulations of a PL-Manifold.-- Publisher description.
Author : Michio Jimbo
Publisher : World Scientific
Page : 728 pages
File Size : 46,7 Mb
Release : 1990-03-01
Category : Science
ISBN : 9789814507066
' This volume will be the first reference book devoted specially to the Yang-Baxter equation. The subject relates to broad areas including solvable models in statistical mechanics, factorized S matrices, quantum inverse scattering method, quantum groups, knot theory and conformal field theory. The articles assembled here cover major works from the pioneering papers to classical Yang-Baxter equation, its quantization, variety of solutions, constructions and recent generalizations to higher genus solutions. Contents:Some Exact Results for the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction (C N Yang)S matrix for the One Dimensional N-Body Problem with Repulsive or Attractive δ-Function Interaction (C N Yang)Partition Function of the Eight-Vertex Lattice Model (R J Baxter)Solutions of the Classical Yang-Baxter Equation and Simple Lie Algebras (A A Belavin & V G Drinfel'd)Some Algebraic Structures Connected with the Yang-Baxter Equation (E K Sklyanin)Quantization of Lie Groups and Lie Algebras (L D Faddeev, N Yu Reshetikhin & L A Takhtajan)Families of Commuting Transfer Matrices in q-State Vertex Models in Non-Linear Integrable Systems — Classical Theory and Quantum Theory (J H H Perk & C L Schultz)Self-Dual Solutions of the Star-Triangle Relations in ZN Models (V A Fateev & A B Zamolodchikov)Solvable Lattice Models Related to the Vector Representation of Classical Simple Lie Algebras (M Jimbo, T Miwa & M Okado)Exactly Solvable SOS Models. II: Proof of the Star-Triangle and Combinatorial Identities (E Date et al.)New Solutions of the Star-Triangle Relations for the Chiral Potts Model (R J Baxter, J H H Perk & H Au-Yang)and other papers Readership: Physicists and mathematicians. Keywords:Yang-Baxter Equation;Star-Triangle Relation;Tetrahedron Equation;R Matrix;Classical R Matrix;Solvable Lattice Model;Factorized Scattering;Quantum Inverse Method;Quantum Groups;Lie Algebra“The collection serves a dual purpose: it provides the physicist or mathematician who works in a different field with an overview of the subject; furthermore, it provides those who work in the subject with a compendium of basic references put conveniently together in one volume.”Mathematical Reveiws “Thus the book gives a good survey of results in one of the hottest points of mathematical physics from the first hands.”Mathematics Abstracts “The second volume is such an excellent, representative collection of articles on the very rich field centered around the Yang-Baxter equation that is should have its place on the shelves of every good library. It is also warmly recommended for people wishing to join this active research area as well as for those who just want to learn the main developments.”Acta Sci. Math. (Szeged) '
Author : Mo-lin Ge
Publisher : World Scientific
Page : 242 pages
File Size : 41,8 Mb
Release : 1992-05-30
Category : Electronic
ISBN : 9789814555838
This volume contains the lectures given by the three speakers, M Jimbo, P P Kulish and E K Sklyanin, who are outstanding experts in their field. It is essential reading to those working in the fields of Quantum Groups, and Integrable Systems.
Author : Chengming Bai,Mo-Lin Ge,Naihuan Jing
Publisher : World Scientific
Page : 215 pages
File Size : 54,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814340458
A note on Brauer-Schur functions / Kazuya Aokage, Hiroshi Mizukawa and Hiro-Fumi Yamada -- [symbol]-operators on associative algebras, associative Yang-Baxter equations and dendriform algebras / Chengming Bai, Li Guo and Xiang Ni -- Irreducible Wakimoto-like modules for the affine Lie algebra [symbol] / Yun Gao and Ziting Zeng -- Verma modules over generic exp-polynomial Lie algebras / Xiangqian Guo, Xuewen Liu and Kaiming Zhao -- A formal infinite dimensional Cauchy problem and its relation to integrable hierarchies / G.F. Helminck, E.A. Panasenko and A.O. Sergeeva -- Partially harmonic tensors and quantized Schur-Weyl duality / Jun Hu and Zhankui Xiao -- Quantum entanglement and approximation by positive matrices / Xiaofen Huang and Naihuan Jing -- 2-partitions of root systems / Bin Li, William Wong and Hechun Zhang -- A survey on weak Hopf algebras / Fang Li and Qinxiu Sun -- The equitable presentation for the quantum algebra Uq(f(k)) / Yan Pan, Meiling Zhu and Libin Li
Author : M A Lifshits,Robert Coquereaux,School on "Quantum symmetries in theoretical physics and mathematics"
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 48,9 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821826553
This volume presents articles from several lectures presented at the school on 'Quantum Symmetries in Theoretical Physics and Mathematics' held in Bariloche, Argentina. The various lecturers provided significantly different points of view on several aspects of Hopf algebras, quantum group theory, and noncommutative differential geometry, ranging from analysis, geometry, and algebra to physical models, especially in connection with integrable systems and conformal field theories. Primary topics discussed in the text include subgroups of quantum $SU(N)$, quantum ADE classifications and generalized Coxeter systems, modular invariance, defects and boundaries in conformal field theory, finite dimensional Hopf algebras, Lie bialgebras and Belavin-Drinfeld triples, real forms of quantum spaces, perturbative and non-perturbative Yang-Baxter operators, braided subfactors in operator algebras and conformal field theory, and generalized ($d^N$) cohomologies.
Author : David E. Radford
Publisher : World Scientific
Page : 584 pages
File Size : 40,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814335997
The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph.
