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Author : Jin Hong,Seok-Jin Kang
Publisher : American Mathematical Soc.
Page : 327 pages
File Size : 47,8 Mb
Release : 2002
Category : Quantum groups
ISBN : 9780821828748
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.
Author : Daniel Bump,Anne Schilling
Publisher : World Scientific Publishing Company
Page : 292 pages
File Size : 47,8 Mb
Release : 2017-01-17
Category : Mathematics
ISBN : 9789814733465
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.
Author : Daniel Bump,Anne Schilling (Mathematician)
Publisher : Unknown
Page : 292 pages
File Size : 42,8 Mb
Release : 2017
Category : Electronic books
ISBN : 9814733458
Author : Jens Carsten Jantzen
Publisher : American Mathematical Soc.
Page : 280 pages
File Size : 50,6 Mb
Release : 2024-06-10
Category : Mathematics
ISBN : 0821872346
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.
Author : George Lusztig
Publisher : Birkhauser
Page : 368 pages
File Size : 49,8 Mb
Release : 1993
Category : Science
ISBN : UOM:39015028905647
Author : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 47,9 Mb
Release : 2016-08-05
Category : Algebraic topology
ISBN : 9781470434410
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.
Author : Masud Chaichian,Andrei Pavlovich Demichev
Publisher : World Scientific
Page : 362 pages
File Size : 44,8 Mb
Release : 1996
Category : Science
ISBN : 9810226233
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.
Author : Christian Kassel
Publisher : Springer Science & Business Media
Page : 540 pages
File Size : 54,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461207832
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.
Author : Anthony Joseph
Publisher : Springer Science & Business Media
Page : 394 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642784002
by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.
Author : Susumu Ariki
Publisher : American Mathematical Soc.
Page : 286 pages
File Size : 42,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853177
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.
Author : Pavel I. Etingof,Olivier Schiffmann
Publisher : Unknown
Page : 242 pages
File Size : 50,9 Mb
Release : 2010
Category : Mathematical physics
ISBN : 1571462074
Author : Shahn Majid
Publisher : Cambridge University Press
Page : 183 pages
File Size : 54,5 Mb
Release : 2002-04-04
Category : Mathematics
ISBN : 9780521010412
Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.
Author : Jonah Blasiak,Ketan D. Mulmuley,Milind Sohoni
Publisher : American Mathematical Soc.
Page : 160 pages
File Size : 45,6 Mb
Release : 2015-04-09
Category : Mathematics
ISBN : 9781470410117
The Kronecker coefficient is the multiplicity of the -irreducible in the restriction of the -irreducible via the natural map , where are -vector spaces and . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. The authors construct two quantum objects for this problem, which they call the nonstandard quantum group and nonstandard Hecke algebra. They show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.
Author : Vyjayanthi Chari,Andrew N. Pressley
Publisher : Cambridge University Press
Page : 672 pages
File Size : 45,7 Mb
Release : 1995-07-27
Category : Mathematics
ISBN : 0521558840
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.
Author : M. Hazewinkel
Publisher : Elsevier
Page : 592 pages
File Size : 47,7 Mb
Release : 2009-07-08
Category : Mathematics
ISBN : 0080932819
Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest. In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc. The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published. A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed. - Thorough and practical source of information - Provides in-depth coverage of new topics in algebra - Includes references to relevant articles, books and lecture notes