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Author: L.A. Lambe
Publisher: Springer Science & Business Media
ISBN: 1461541093
Category : Mathematics
Languages : en
Pages : 314
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Book Description
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: L.A. Lambe
Publisher: Springer Science & Business Media
ISBN: 1461541093
Category : Mathematics
Languages : en
Pages : 314
Get Book
Book Description
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
ISBN: 9781139437028
Category : Mathematics
Languages : en
Pages : 246
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Book Description
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
Publisher: Oxford University Press on Demand
ISBN: 0198530684
Category : Mathematics
Languages : en
Pages : 151
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Book Description
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: Michio Jimbo
Publisher: World Scientific
ISBN: 9789810201210
Category : Science
Languages : en
Pages : 740
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Book Description
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.
Author: L C Biedenharn
Publisher: World Scientific
ISBN: 9814500135
Category : Science
Languages : en
Pages : 304
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Book Description
Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction. Tensor operators are defined and discussed with examples, and a systematic treatment of the important (3j) series of operators is developed in detail. This book is a good reference for graduate students in physics and mathematics. Contents:Origins of Quantum GroupsRepresentations of Unitary Quantum GroupsTensor Operators in Quantum GroupsThe Dual Algebra and the Factor GroupQuantum Rotation MatricesQuantum Groups at Roots of UnityAlgebraic Induction of Quantum Group RepresentationsSpecial TopicsBibliographyIndex Readership: Physicists and mathematicians interested in symmetry techniques in physics. keywords:Quantum Groups;Quantum Algebras;Tensor Operators;Symmetries;Representations;q-Boson Operators;q-Clebsch-Gordan Coefficients;Vector Coherent States;Algebraic Induction;Weyl-Ordered Polynomials
Author: Christian Kassel
Publisher: Springer
ISBN:
Category : Mathematics
Languages : en
Pages : 560
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Book Description
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: Mo-lin Ge
Publisher: World Scientific
ISBN: 9814555835
Category :
Languages : en
Pages : 242
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Book Description
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: Shahn Majid
Publisher: Cambridge University Press
ISBN: 0521010411
Category : Mathematics
Languages : en
Pages : 183
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Book Description
Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.
Author: Florin Felix Nichita
Publisher: MDPI
ISBN: 3038973246
Category : Mathematics
Languages : en
Pages : 239
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Book Description
This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms
Author: Vladimir K. Dobrev
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110427788
Category : Science
Languages : en
Pages : 406
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Book Description
With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups. Contents Quantum Groups and Quantum Algebras Highest-Weight Modules over Quantum Algebras Positive-Energy Representations of Noncompact Quantum Algebras Duality for Quantum Groups Invariant q-Difference Operators Invariant q-Difference Operators Related to GLq(n) q-Maxwell Equations Hierarchies
