Quantum Group Symmetry and Q-tensor Algebras
Quantum Group Symmetry and Q-tensor Algebras

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.

Quantum Groups
Quantum Groups

Author: Vladimir K. Dobrev

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2017-07-10

Total Pages: 450

ISBN-13: 3110427788

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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

Tensor Categories
Tensor Categories

Author: Pavel Etingof

Publisher: American Mathematical Soc.

Published: 2016-08-05

Total Pages: 362

ISBN-13: 1470434415

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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.

Quantum Symmetries
Quantum Symmetries

Author: Guillaume Aubrun

Publisher: Springer

Published: 2017-10-11

Total Pages: 126

ISBN-13: 331963206X

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Providing an introduction to current research topics in functional analysis and its applications to quantum physics, this book presents three lectures surveying recent progress and open problems. A special focus is given to the role of symmetry in non-commutative probability, in the theory of quantum groups, and in quantum physics. The first lecture presents the close connection between distributional symmetries and independence properties. The second introduces many structures (graphs, C*-algebras, discrete groups) whose quantum symmetries are much richer than their classical symmetry groups, and describes the associated quantum symmetry groups. The last lecture shows how functional analytic and geometric ideas can be used to detect and to quantify entanglement in high dimensions. The book will allow graduate students and young researchers to gain a better understanding of free probability, the theory of compact quantum groups, and applications of the theory of Banach spaces to quantum information. The latter applications will also be of interest to theoretical and mathematical physicists working in quantum theory.

Foundations of Quantum Group Theory
Foundations of Quantum Group Theory

Author: Shahn Majid

Publisher: Cambridge University Press

Published: 2000

Total Pages: 668

ISBN-13: 9780521648684

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A graduate level text which systematically lays out the foundations of Quantum Groups.

Factorizable Sheaves and Quantum Groups
Factorizable Sheaves and Quantum Groups

Author: Roman Bezrukavnikov

Publisher: Springer

Published: 2006-11-14

Total Pages: 300

ISBN-13: 3540692312

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The book is devoted to the geometrical construction of the representations of Lusztig's small quantum groups at roots of unity. These representations are realized as some spaces of vanishing cycles of perverse sheaves over configuration spaces. As an application, the bundles of conformal blocks over the moduli spaces of curves are studied. The book is intended for specialists in group representations and algebraic geometry.

A Quantum Groups Primer
A Quantum Groups Primer

Author: Shahn Majid

Publisher: Cambridge University Press

Published: 2002-04-04

Total Pages: 183

ISBN-13: 0521010411

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

A Guide to Quantum Groups
A Guide to Quantum Groups

Author: Vyjayanthi Chari

Publisher: Cambridge University Press

Published: 1995-07-27

Total Pages: 672

ISBN-13: 9780521558846

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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.