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Author: Jin Hong Publisher: American Mathematical Soc. ISBN: 0821828746 Category : Mathematics Languages : en Pages : 327
Book Description
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: Jin Hong Publisher: American Mathematical Soc. ISBN: 0821828746 Category : Mathematics Languages : en Pages : 327
Book Description
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 Publisher: World Scientific Publishing Company ISBN: 9814733466 Category : Mathematics Languages : en Pages : 292
Book Description
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 Publisher: World Scientific Publishing Company ISBN: 9789814733434 Category : Combinatorial analysis Languages : en Pages : 0
Book Description
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 the ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear group, and phenomena in combinatorics. The authors are both contributors to Sage, an open-source mathematical software system, which has strong support for crystal bases and combinatorics and the book takes advantage of this.
Author: George Lusztig Publisher: Springer Science & Business Media ISBN: 0817647171 Category : Mathematics Languages : en Pages : 361
Book Description
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.
Author: Anatoli Klimyk Publisher: Springer Science & Business Media ISBN: 3642608965 Category : Science Languages : en Pages : 568
Book Description
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.
Author: Jens Carsten Jantzen Publisher: American Mathematical Soc. ISBN: 0821804782 Category : Mathematics Languages : en Pages : 282
Book Description
The material is very well motivated ... Of the various monographs available on quantum groups, this one ... seems the most suitable for most mathematicians new to the subject ... will also be appreciated by a lot of those with considerably more experience. --Bulletin of the London Mathematical Society Since its origin, the theory of quantum groups has become one of the most fascinating topics of modern mathematics, with numerous applications to several sometimes rather disparate areas, including low-dimensional topology and mathematical physics. This book is one of the first expositions that is specifically directed to students who have no previous knowledge of the subject. The only prerequisite, in addition to standard linear algebra, is some acquaintance with the classical theory of complex semisimple Lie algebras. Starting with the quantum analog of $\mathfrak{sl}_2$, 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 algebras. The choice of the topics and the style of exposition make Jantzen's book an excellent textbook for a one-semester course on quantum groups.
Author: Pavel Etingof Publisher: American Mathematical Soc. ISBN: 1470434415 Category : Mathematics Languages : en Pages : 362
Book Description
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: Anthony Joseph Publisher: Springer Science & Business Media ISBN: 3642784003 Category : Mathematics Languages : en Pages : 394
Book Description
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: Jin Hong
Author: Jin Hong
Author: Daniel Bump
Author: Daniel Bump
Author: George Lusztig
Author: Anatoli Klimyk
Author: Jens Carsten Jantzen
Author: Pavel Etingof
Author: Dijana Jakelić
Author: Dijana Jakelić
Author: Anthony Joseph