Quantum Groups And Their Representations

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Quantum Groups and Their Representations

Anatoli Klimyk,Konrad Schmüdgen
Quantum Groups and Their Representations Book PDF

Author : Anatoli Klimyk,Konrad Schmüdgen
Publisher : Springer Science & Business Media
Page : 568 pages
File Size : 51,5 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642608964

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Quantum Groups and Their Representations by Anatoli Klimyk,Konrad Schmüdgen Pdf

This book start with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and in mathematics. It can also be used as a reference by more advanced readers. The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The book is written with potential applications in physics and mathematics in mind. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. A number of topics and results from the more advanced general theory are developed and discussed.

Quantum Theory, Groups and Representations

Peter Woit
Quantum Theory Groups and Representations Book PDF

Author : Peter Woit
Publisher : Springer
Page : 668 pages
File Size : 46,6 Mb
Release : 2017-11-01
Category : Science
ISBN : 9783319646121

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Quantum Theory, Groups and Representations by Peter Woit Pdf

This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.

A Guide to Quantum Groups

Vyjayanthi Chari,Andrew N. Pressley
A Guide to Quantum Groups Book PDF

Author : Vyjayanthi Chari,Andrew N. Pressley
Publisher : Cambridge University Press
Page : 672 pages
File Size : 43,6 Mb
Release : 1995-07-27
Category : Mathematics
ISBN : 0521558840

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A Guide to Quantum Groups by Vyjayanthi Chari,Andrew N. Pressley Pdf

Since they first arose in the 1970s and early 1980s, quantum groups have proved to be of great interest to mathematicians and theoretical physicists. The theory of quantum groups is now well established as a fascinating chapter of representation theory, and has thrown new light on many different topics, notably low-dimensional topology and conformal field theory. The goal of this book is to give a comprehensive view of quantum groups and their applications. The authors build on a self-contained account of the foundations of the subject and go on to treat the more advanced aspects concisely and with detailed references to the literature. Thus this book can serve both as an introduction for the newcomer, and as a guide for the more experienced reader. All who have an interest in the subject will welcome this unique treatment of quantum groups.

Lectures on Algebraic Quantum Groups

Ken Brown,Ken R. Goodearl
Lectures on Algebraic Quantum Groups Book PDF

Author : Ken Brown,Ken R. Goodearl
Publisher : Birkhäuser
Page : 339 pages
File Size : 52,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034882057

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Lectures on Algebraic Quantum Groups by Ken Brown,Ken R. Goodearl Pdf

This book consists of an expanded set of lectures on algebraic aspects of quantum groups. It particularly concentrates on quantized coordinate rings of algebraic groups and spaces and on quantized enveloping algebras of semisimple Lie algebras. Large parts of the material are developed in full textbook style, featuring many examples and numerous exercises; other portions are discussed with sketches of proofs, while still other material is quoted without proof.

Algebras of Functions on Quantum Groups: Part I

Leonid I. Korogodski,Leonid I.. Korogodski,Yan S. Soibelman,Yan S.. Soibelman
Algebras of Functions on Quantum Groups Part I Book PDF

Author : Leonid I. Korogodski,Leonid I.. Korogodski,Yan S. Soibelman,Yan S.. Soibelman
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 49,8 Mb
Release : 1998
Category : Function algebras
ISBN : 9780821803363

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Algebras of Functions on Quantum Groups: Part I by Leonid I. Korogodski,Leonid I.. Korogodski,Yan S. Soibelman,Yan S.. Soibelman Pdf

The text is devoted to the study of algebras of functions on quantum groups. The book includes the theory of Poisson-Lie algebras (quasi-classical version of algebras of functions on quantum groups), a description of representations of algebras of functions and the theory of quantum Weyl groups. It can serve as a text for an introduction to the theory of quantum groups and is intended for graduate students and research mathematicians working in algebra, representation theory and mathematical physics.

Representation Theory of Algebraic Groups and Quantum Groups

Toshiaki Shoji
Representation Theory of Algebraic Groups and Quantum Groups Book PDF

Author : Toshiaki Shoji
Publisher : American Mathematical Society(RI)
Page : 514 pages
File Size : 42,6 Mb
Release : 2004
Category : Computers
ISBN : UOM:39015061859339

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Representation Theory of Algebraic Groups and Quantum Groups by Toshiaki Shoji Pdf

A collection of research and survey papers written by speakers at the Mathematical Society of Japan's 10th International Conference. This title presents an overview of developments in representation theory of algebraic groups and quantum groups. It includes papers containing results concerning Lusztig's conjecture on cells in affine Weyl groups.

Quantum Groups, Quantum Categories and Quantum Field Theory

Jürg Fröhlich,Thomas Kerler
Quantum Groups Quantum Categories and Quantum Field Theory Book PDF

Author : Jürg Fröhlich,Thomas Kerler
Publisher : Springer
Page : 438 pages
File Size : 45,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540476115

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Quantum Groups, Quantum Categories and Quantum Field Theory by Jürg Fröhlich,Thomas Kerler Pdf

This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.

Elliptic Quantum Groups

Hitoshi Konno
Elliptic Quantum Groups Book PDF

Author : Hitoshi Konno
Publisher : Springer Nature
Page : 139 pages
File Size : 40,7 Mb
Release : 2020-09-14
Category : Science
ISBN : 9789811573873

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Elliptic Quantum Groups by Hitoshi Konno Pdf

This is the first book on elliptic quantum groups, i.e., quantum groups associated to elliptic solutions of the Yang-Baxter equation. Based on research by the author and his collaborators, the book presents a comprehensive survey on the subject including a brief history of formulations and applications, a detailed formulation of the elliptic quantum group in the Drinfeld realization, explicit construction of both finite and infinite-dimensional representations, and a construction of the vertex operators as intertwining operators of these representations. The vertex operators are important objects in representation theory of quantum groups. In this book, they are used to derive the elliptic q-KZ equations and their elliptic hypergeometric integral solutions. In particular, the so-called elliptic weight functions appear in such solutions. The author’s recent study showed that these elliptic weight functions are identified with Okounkov’s elliptic stable envelopes for certain equivariant elliptic cohomology and play an important role to construct geometric representations of elliptic quantum groups. Okounkov’s geometric approach to quantum integrable systems is a rapidly growing topic in mathematical physics related to the Bethe ansatz, the Alday-Gaiotto-Tachikawa correspondence between 4D SUSY gauge theories and the CFT’s, and the Nekrasov-Shatashvili correspondences between quantum integrable systems and quantum cohomology. To invite the reader to such topics is one of the aims of this book.

Quantum Groups

Ross Street
Quantum Groups Book PDF

Author : Ross Street
Publisher : Cambridge University Press
Page : 160 pages
File Size : 48,6 Mb
Release : 2007-01-18
Category : Mathematics
ISBN : 9781139461443

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Quantum Groups by Ross Street Pdf

Algebra has moved well beyond the topics discussed in standard undergraduate texts on 'modern algebra'. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an 'algebra'. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a 'coalgebra'. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term 'quantum group', along with revolutionary new examples, was launched by Drinfel'd in 1986.

Quantum Group Symmetry and Q-tensor Algebras

L. C. Biedenharn,M. A. Lohe
Quantum Group Symmetry and Q tensor Algebras Book PDF

Author : L. C. Biedenharn,M. A. Lohe
Publisher : World Scientific
Page : 305 pages
File Size : 54,9 Mb
Release : 1995
Category : Science
ISBN : 9789810223311

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Quantum Group Symmetry and Q-tensor Algebras by L. C. Biedenharn,M. A. Lohe Pdf

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.

Introduction to Quantum Groups

George Lusztig
Introduction to Quantum Groups Book PDF

Author : George Lusztig
Publisher : Springer Science & Business Media
Page : 352 pages
File Size : 47,6 Mb
Release : 2010-10-27
Category : Mathematics
ISBN : 9780817647179

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Introduction to Quantum Groups by George Lusztig Pdf

The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical basis with rather remarkable properties. This book will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists and to theoretical physicists and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used as a text book.

Tensor Categories

Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Tensor Categories Book PDF

Author : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 42,9 Mb
Release : 2016-08-05
Category : Algebraic topology
ISBN : 9781470434410

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Tensor Categories by Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik Pdf

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Complex Semisimple Quantum Groups and Representation Theory

Christian Voigt,Robert Yuncken
Complex Semisimple Quantum Groups and Representation Theory Book PDF

Author : Christian Voigt,Robert Yuncken
Publisher : Springer Nature
Page : 382 pages
File Size : 50,5 Mb
Release : 2020-09-24
Category : Mathematics
ISBN : 9783030524630

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Complex Semisimple Quantum Groups and Representation Theory by Christian Voigt,Robert Yuncken Pdf

This book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group. The main components are: - a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism, - the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals, - algebraic representation theory in terms of category O, and - analytic representation theory of quantized complex semisimple groups. Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.

A Quantum Groups Primer

Shahn Majid
A Quantum Groups Primer Book PDF

Author : Shahn Majid
Publisher : Cambridge University Press
Page : 183 pages
File Size : 41,8 Mb
Release : 2002-04-04
Category : Mathematics
ISBN : 9780521010412

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A Quantum Groups Primer by Shahn Majid Pdf

Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.

Quantum Groups

Vladimir K. Dobrev
Quantum Groups Book PDF

Author : Vladimir K. Dobrev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 406 pages
File Size : 44,7 Mb
Release : 2017-07-10
Category : Science
ISBN : 9783110427783

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Quantum Groups by Vladimir K. Dobrev Pdf

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