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 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.
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.
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.
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.
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.
Complex Semisimple Quantum Groups And Representation Theory
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Author : Christian Voigt
language : en
Publisher: Springer Nature
Release Date : 2020-09-24
Complex Semisimple Quantum Groups And Representation Theory written by Christian Voigt and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-24 with Mathematics categories.
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.
Representation Theory Of Algebraic Groups And Quantum Groups
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Author : Akihiko Gyoja
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-11-25
Representation Theory Of Algebraic Groups And Quantum Groups written by Akihiko Gyoja 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 2010-11-25 with Mathematics categories.
Invited articles by top notch experts Focus is on topics in representation theory of algebraic groups and quantum groups Of interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics
Representations Of Algebraic Groups Quantum Groups And Lie Algebras
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Author : Georgia Benkart
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Representations Of Algebraic Groups Quantum Groups And Lie Algebras written by Georgia Benkart 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 2006 with Mathematics categories.
Covers various aspects of the representation theory of Lie algebras, finite groups of Lie types, Hecke algebras, and Lie super algebras. This book outlines connections among irreducible representations of certain blocks of reduced enveloping algebras of semi-simple Lie algebras in positive characteristic.
Geometric Representation Theory And Extended Affine Lie Algebras
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Author : Erhard Neher
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
Geometric Representation Theory And Extended Affine Lie Algebras written by Erhard Neher 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 2011 with Mathematics categories.
Lie theory has connections to many other disciplines such as geometry, number theory, mathematical physics, and algebraic combinatorics. The interaction between algebra, geometry and combinatorics has proven to be extremely powerful in shedding new light on each of these areas. This book presents the lectures given at the Fields Institute Summer School on Geometric Representation Theory and Extended Affine Lie Algebras held at the University of Ottawa in 2009. It provides a systematic account by experts of some of the exciting developments in Lie algebras and representation theory in the last two decades. It includes topics such as geometric realizations of irreducible representations in three different approaches, combinatorics and geometry of canonical and crystal bases, finite $W$-algebras arising as the quantization of the transversal slice to a nilpotent orbit, structure theory of extended affine Lie algebras, and representation theory of affine Lie algebras at level zero. This book will be of interest to mathematicians working in Lie algebras and to graduate students interested in learning the basic ideas of some very active research directions. The extensive references in the book will be helpful to guide non-experts to the original sources.
Algebraic Groups And Quantum Groups
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Author : Susumu Ariki
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Algebraic Groups And Quantum Groups written by Susumu Ariki 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 2012 with Mathematics categories.
This volume contains the proceedings of the tenth international conference on Representation Theory of Algebraic Groups and Quantum Groups, held August 2-6, 2010, at Nagoya University, Nagoya, Japan. The survey articles and original papers contained in this volume offer a comprehensive view of current developments in the field. Among others reflecting recent trends, one central theme is research on representations in the affine case. In three articles, the authors study representations of W-algebras and affine Lie algebras at the critical level, and three other articles are related to crystals in the affine case, that is, Mirkovic-Vilonen polytopes for affine type $A$ and Kerov-Kirillov-Reshetikhin type bijection for affine type $E_6$. Other contributions cover a variety of topics such as modular representation theory of finite groups of Lie type, quantum queer super Lie algebras, Khovanov's arc algebra, Hecke algebras and cyclotomic $q$-Schur algebras, $G_1T$-Verma modules for reductive algebraic groups, equivariant $K$-theory of quantum vector bundles, and the cluster algebra. This book is suitable for graduate students and researchers interested in geometric and combinatorial representation theory, and other related fields.