Finite Dimensional Algebras And Quantum Groups

Author : Bangming Deng
Publisher : American Mathematical Soc.
Page : 790 pages
File Size : 40,9 Mb
Release : 2008
Category : Finite fields (Algebra).
ISBN : 9780821841860

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Finite Dimensional Algebras and Quantum Groups by Bangming Deng Pdf

"The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak{gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature."--Publisher's website.

Author : Vlastimil Dlab,Claus Michael Ringel
Publisher : American Mathematical Soc.
Page : 502 pages
File Size : 55,7 Mb
Release : 2004
Category : Associative algebras
ISBN : 9780821834169

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Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry by Vlastimil Dlab,Claus Michael Ringel Pdf

These proceedings are from the Tenth International Conference on Representations of Algebras and Related Topics (ICRA X) held at The Fields Institute. In addition to the traditional ``instructional'' workshop preceding the conference, there were also workshops on ``Commutative Algebra, Algebraic Geometry and Representation Theory'', ``Finite Dimensional Algebras, Algebraic Groups and Lie Theory'', and ``Quantum Groups and Hall Algebras''. These workshops reflect the latest developments and the increasing interest in areas that are closely related to the representation theory of finite dimensional associative algebras. Although these workshops were organized separately, their topics are strongly interrelated. The workshop on Commutative Algebra, Algebraic Geometry and Representation Theory surveyed various recently established connections, such as those pertaining to the classification of vector bundles or Cohen-Macaulay modules over Noetherian rings, coherent sheaves on curves, or ideals in Weyl algebras. In addition, methods from algebraic geometry or commutative algebra relating to quiver representations and varieties of modules were presented. The workshop on Finite Dimensional Algebras, Algebraic Groups and Lie Theory surveyed developments in finite dimensional algebras and infinite dimensional Lie theory, especially as the two areas interact and may have future interactions. The workshop on Quantum Groups and Hall Algebras dealt with the different approaches of using the representation theory of quivers (and species) in order to construct quantum groups, working either over finite fields or over the complex numbers. In particular, these proceedings contain a quite detailed outline of the use of perverse sheaves in order to obtain canonical bases. The book is recommended for graduate students and researchers in algebra and geometry.

Author : Anatoli Klimyk,Konrad Schmüdgen
Publisher : Springer Science & Business Media
Page : 568 pages
File Size : 49,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642608964

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Quantum Groups and Their Representations by Anatoli Klimyk,Konrad Schmüdgen Pdf

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 : Leonid I. Korogodski,Leonid I.. Korogodski,Yan S. Soibelman,Yan S.. Soibelman
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 44,8 Mb
Release : 1998
Category : Function algebras
ISBN : 9780821803363

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Algebras of Functions on Quantum Groups: Part I by Leonid I. Korogodski,Leonid I.. Korogodski,Yan S. Soibelman,Yan S.. Soibelman Pdf

The text is devoted to the study of algebras of functions on quantum groups. The book includes the theory of Poisson-Lie algebras (quasi-classical version of algebras of functions on quantum groups), a description of representations of algebras of functions and the theory of quantum Weyl groups. It can serve as a text for an introduction to the theory of quantum groups and is intended for graduate students and research mathematicians working in algebra, representation theory and mathematical physics.

Author : Ken Brown,Ken R. Goodearl
Publisher : Birkhäuser
Page : 339 pages
File Size : 49,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034882057

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Lectures on Algebraic Quantum Groups by Ken Brown,Ken R. Goodearl Pdf

This book consists of an expanded set of lectures on algebraic aspects of quantum groups. It particularly concentrates on quantized coordinate rings of algebraic groups and spaces and on quantized enveloping algebras of semisimple Lie algebras. Large parts of the material are developed in full textbook style, featuring many examples and numerous exercises; other portions are discussed with sketches of proofs, while still other material is quoted without proof.

Author : Andrew Pressley
Publisher : Cambridge University Press
Page : 246 pages
File Size : 46,5 Mb
Release : 2002-01-17
Category : Mathematics
ISBN : 113943702X

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Quantum Groups and Lie Theory by Andrew Pressley Pdf

This book comprises an overview of the material presented at the 1999 Durham Symposium on Quantum Groups and includes contributions from many of the world's leading figures in this area. It will be of interest to researchers and will also be useful as a reference text for graduate courses.

Author : Benjamin Enriquez
Publisher : European Mathematical Society
Page : 148 pages
File Size : 52,8 Mb
Release : 2008
Category : Mathematics
ISBN : 3037190477

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Quantum Groups by Benjamin Enriquez Pdf

The volume starts with a lecture course by P. Etingof on tensor categories (notes by D. Calaque). This course is an introduction to tensor categories, leading to topics of recent research such as realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf algebras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius-Perron dimensions, and the classification of tensor categories. The remainder of the book consists of three detailed expositions on associators and the Vassiliev invariants of knots, classical and quantum integrable systems and elliptic algebras, and the groups of algebra automorphisms of quantum groups. The preface puts the results presented in perspective. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume gives an overview of the ongoing research in the domain of quantum groups, an important subject of current mathematical physics.

Author : Anthony Joseph
Publisher : Springer Science & Business Media
Page : 394 pages
File Size : 55,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642784002

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Quantum Groups and Their Primitive Ideals by Anthony Joseph Pdf

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 : Edward L. Green,Birge Huisgen-Zimmermann
Publisher : American Mathematical Soc.
Page : 356 pages
File Size : 44,6 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821809280

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Trends in the Representation Theory of Finite Dimensional Algebras by Edward L. Green,Birge Huisgen-Zimmermann Pdf

This refereed collection of research papers and survey articles reflects the interplay of finite-dimensional algebras with other areas (algebraic geometry, homological algebra, and the theory of quantum groups). Current trends are presented from the discussions at the AMS-IMS-SIAM Joint Summer Research Conference at the University of Washington (Seattle). The volume features several excellent expository articles which will introduce the beginning researcher to cutting-edge topics in representation theory. The book will also provide inspiration to researchers in related areas, as it includes original papers spanning a broad spectrum of representation theory. It's features include: work outlining significant progress on long-standing open problems; survey articles offering both overviews and introductions to various subfields of the topic; and, expositions reflecting the interplay between the representation theory of algebras and other fields.

Author : Vyjayanthi Chari,Andrew N. Pressley
Publisher : Cambridge University Press
Page : 672 pages
File Size : 45,8 Mb
Release : 1995-07-27
Category : Mathematics
ISBN : 0521558840

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A Guide to Quantum Groups by Vyjayanthi Chari,Andrew N. Pressley Pdf

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 : Shahn Majid
Publisher : Cambridge University Press
Page : 668 pages
File Size : 47,9 Mb
Release : 2000
Category : Group theory
ISBN : 0521648688

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Foundations of Quantum Group Theory by Shahn Majid Pdf

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

Author : William Joseph Haboush Brian Parshall
Publisher : American Mathematical Soc.
Page : 442 pages
File Size : 40,7 Mb
Release : 1994-05-02
Category : Linear algebraic groups
ISBN : 0821868284

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Algebraic groups and their generalizations: Quantum and infinite-dimensional methods by William Joseph Haboush Brian Parshall Pdf

Author : Bangming Deng,Jie Du,Qiang Fu
Publisher : Cambridge University Press
Page : 217 pages
File Size : 41,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781107608603

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A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Bangming Deng,Jie Du,Qiang Fu Pdf

The first book of its kind to present an algebraic approach to affine q-Schur algebras and affine quantum Schur-Weyl theory.

Author : Florin Felix Nichita
Publisher : MDPI
Page : 239 pages
File Size : 54,7 Mb
Release : 2019-01-31
Category : Mathematics
ISBN : 9783038973249

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Hopf Algebras, Quantum Groups and Yang-Baxter Equations by Florin Felix Nichita Pdf

This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms

Author : Johnny T. Ottesen
Publisher : Springer Science & Business Media
Page : 223 pages
File Size : 49,6 Mb
Release : 1995-04-18
Category : Science
ISBN : 9783540589143

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Infinite Dimensional Groups and Algebras in Quantum Physics by Johnny T. Ottesen Pdf

The idea of writing this book appeared when I was working on some problems related to representations of physically relevant infinite - mensional groups of operators on physically relevant Hilbert spaces. The considerations were local, reducing the subject to dealing with representations of infinite-dimensional Lie algebras associated with the associated groups. There is a large number of specialized articles and books on parts of this subject, but to our suprise only a few represent the point of view given in this book. Moreover, none of the written material was self-contained. At present, the subject has not reached its final form and active research is still being undertaken. I present this subject of growing importance in a unified manner and by a fairly simple approach. I present a route by which students can absorb and understand the subject, only assuming that the reader is familliar with functional analysis, especially bounded and unbounded operators on Hilbert spaces. Moreover, I assume a little basic knowledge of algebras , Lie algebras, Lie groups, and manifolds- at least the definitions. The contents are presented in detail in the introduction in Chap. The manuscript of this book has been succesfully used by some advanced graduate students at Aarhus University, Denmark, in their "A-exame'. I thank them for comments.