Algebras of Functions on Quantum Groups: Part I

Algebras of Functions on Quantum Groups: Part I PDF

Author: Leonid I. Korogodski

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 162

ISBN-13: 0821803360

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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.

Quantum Groups and Their Representations

Quantum Groups and Their Representations PDF

Author: Anatoli Klimyk

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 568

ISBN-13: 3642608965

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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.

Introduction to Quantum Groups

Introduction to Quantum Groups PDF

Author: George Lusztig

Publisher: Springer Science & Business Media

Published: 2010-10-27

Total Pages: 361

ISBN-13: 0817647171

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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.

Quantum Group Symmetry and Q-tensor Algebras

Quantum Group Symmetry and Q-tensor Algebras PDF

Author: L. C. Biedenharn

Publisher: World Scientific

Published: 1995

Total Pages: 305

ISBN-13: 9810223315

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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.

An Invitation to Quantum Groups and Duality

An Invitation to Quantum Groups and Duality PDF

Author: Thomas Timmermann

Publisher: European Mathematical Society

Published: 2008

Total Pages: 436

ISBN-13: 9783037190432

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This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. Part I of the text presents the basic theory of Hopf algebras, Van Daele's duality theory of algebraic quantum groups, and Woronowicz's compact quantum groups, staying in a purely algebraic setting. Part II focuses on quantum groups in the setting of operator algebras. Woronowicz's compact quantum groups are treated in the setting of $C^*$-algebras, and the fundamental multiplicative unitaries of Baaj and Skandalis are studied in detail. An outline of Kustermans' and Vaes' comprehensive theory of locally compact quantum groups completes this part. Part III leads to selected topics, such as coactions, Baaj-Skandalis-duality, and approaches to quantum groupoids in the setting of operator algebras. The book is addressed to graduate students and non-experts from other fields. Only basic knowledge of (multi-) linear algebra is required for the first part, while the second and third part assume some familiarity with Hilbert spaces, $C^*$-algebras, and von Neumann algebras.

Introduction to Quantum Groups

Introduction to Quantum Groups PDF

Author: Masud Chaichian

Publisher: World Scientific

Published: 1996

Total Pages: 362

ISBN-13: 9789810226237

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In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system — harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincaré symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.

Algebraic Combinatorics And Quantum Groups

Algebraic Combinatorics And Quantum Groups PDF

Author: Naihuan Jing

Publisher: World Scientific

Published: 2003-06-27

Total Pages: 171

ISBN-13: 9814485500

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Algebraic combinatorics has evolved into one of the most active areas of mathematics during the last several decades. Its recent developments have become more interactive with not only its traditional field representation theory but also algebraic geometry, harmonic analysis and mathematical physics.This book presents articles from some of the key contributors in the area. It covers Hecke algebras, Hall algebras, the Macdonald polynomial and its deviations, and their relations with other fields.

Quantum Groups

Quantum Groups PDF

Author: Christian Kassel

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 540

ISBN-13: 1461207835

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Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.

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