Introduction To Quantum Groups And Crystal Bases
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Introduction To Quantum Groups And Crystal Bases
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Author : Jin Hong
language : en
Publisher: American Mathematical Society
Release Date : 2025-02-06
Introduction To Quantum Groups And Crystal Bases written by Jin Hong and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-02-06 with Mathematics categories.
The notion of a “quantum group” was introduced by V.G. Drinfel'd and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of Kac-Moody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$-algebras. In particular, the theory of “crystal bases” or “canonical bases” developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.
Introduction To Quantum Groups And Crystal Bases
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Author : Jin Hong
language : en
Publisher: American Mathematical Soc.
Release Date : 2002
Introduction To Quantum Groups And Crystal Bases written by Jin Hong and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.
Crystal Bases Representations And Combinatorics
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Author : Daniel Bump
language : en
Publisher: World Scientific Publishing Company
Release Date : 2017-01-17
Crystal Bases Representations And Combinatorics written by Daniel Bump and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-17 with Mathematics categories.
This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.
Introduction To Quantum Groups
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Author : George Lusztig
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-27
Introduction To Quantum Groups written by George Lusztig and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-27 with Mathematics categories.
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
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Author : Pavel Etingof
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-08-05
Tensor Categories written by Pavel Etingof and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-05 with Mathematics categories.
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.
Lectures On Quantum Groups
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Author : Pavel I. Etingof
language : en
Publisher:
Release Date : 2002
Lectures On Quantum Groups written by Pavel I. Etingof and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematical physics categories.
Based on lectures given at Harvard University in 1997, this book is an introduction to the theory of quantum groups and its development between 1982 and 1997. Topics covered include: relevant quasiclassical objects; bialgebras; Hopf algebras; and lie associators.
Lectures On Quantum Groups
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Author : Jens Carsten Jantzen
language : en
Publisher: American Mathematical Soc.
Release Date :
Lectures On Quantum Groups written by Jens Carsten Jantzen and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
Starting with the quantum analog of sl2, the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebra.
A Guide To Quantum Groups
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Author : Vyjayanthi Chari
language : en
Publisher: Cambridge University Press
Release Date : 1995-07-27
A Guide To Quantum Groups written by Vyjayanthi Chari and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-07-27 with Mathematics categories.
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.
Symmetry
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Author : Roy McWeeny
language : en
Publisher: Courier Corporation
Release Date : 2012-05-23
Symmetry written by Roy McWeeny and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-23 with Science categories.
Well-organized volume develops ideas of group and representation theory in progressive fashion. Emphasis on finite groups describing symmetry of regular polyhedra and of repeating patterns, plus geometric illustrations.