Identities Of Algebras And Their Representations

Author : I︠U︡riĭ Pitrimovich Razmyslov
Publisher : American Mathematical Soc.
Page : 468 pages
File Size : 54,8 Mb
Release : 1994
Category : Mathematics
ISBN : 0821846086

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Identities of Algebras and their Representations by I︠U︡riĭ Pitrimovich Razmyslov Pdf

During the past forty years, a new trend in the theory of associative algebras, Lie algebras, and their representations has formed under the influence of mathematical logic and universal algebra, namely, the theory of varieties and identities of associative algebras, Lie algebras, and their representations. The last twenty years have seen the creation of the method of 2-words and *a-functions, which allowed a number of problems in the theory of groups, rings, Lie algebras, and their representations to be solved in a unified way. The possibilities of this method are far from exhausted. This book sums up the applications of the method of 2-words and *a-functions in the theory of varieties and gives a systematic exposition of contemporary achievements in the theory of identities of algebras and their representations closely related to this method. The aim is to make these topics accessible to a wider group of mathematicians.

Author : Eli Aljadeff,Antonio Giambruno,Claudio Procesi,Amitai Regev
Publisher : American Mathematical Soc.
Page : 630 pages
File Size : 40,7 Mb
Release : 2020-12-14
Category : Education
ISBN : 9781470451745

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Rings with Polynomial Identities and Finite Dimensional Representations of Algebras by Eli Aljadeff,Antonio Giambruno,Claudio Procesi,Amitai Regev Pdf

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

Author : Onofrio Mario Di Vincenzo,Antonio Giambruno
Publisher : Springer Nature
Page : 421 pages
File Size : 48,9 Mb
Release : 2021-03-22
Category : Mathematics
ISBN : 9783030631116

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Polynomial Identities in Algebras by Onofrio Mario Di Vincenzo,Antonio Giambruno Pdf

This volume contains the talks given at the INDAM workshop entitled "Polynomial identites in algebras", held in Rome in September 2019. The purpose of the book is to present the current state of the art in the theory of PI-algebras. The review of the classical results in the last few years has pointed out new perspectives for the development of the theory. In particular, the contributions emphasize on the computational and combinatorial aspects of the theory, its connection with invariant theory, representation theory, growth problems. It is addressed to researchers in the field.

Author : V. N. Gerasimov,N. G. Nesterenko,A. I. Valitskas
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 48,7 Mb
Release : 1993
Category : Mathematics
ISBN : 0821875035

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Three Papers on Algebras and Their Representations by V. N. Gerasimov,N. G. Nesterenko,A. I. Valitskas Pdf

This book contains the doctoral dissertations of three students from Novosibirsk who participated in the seminar of L. A. Bokut'. The dissertation of Gerasimov focuses on Cohn's theory of noncommutative matrix localizations. Gerasimov presents a construction of matrix localization that is not directly related to (prime) matrix ideals of Cohn, but rather deals with localizations of arbitrary subsets of matrices over a ring. The work of Valitskas applies ideas and constructions of Gerasimov to embeddings of rings into radical rings (in the sense of Jacobson) to develop a theory essentially parallel to Cohn's theory of embeddings of rings into skew fields. Nesterenko's dissertation solves some important problems of Anan'in and Bergman about representations of (infinite-dimensional) algebras and categories in (triangular) matrices over commutative rings.

Author : James Lepowsky,Haisheng Li
Publisher : Springer Science & Business Media
Page : 330 pages
File Size : 47,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817681869

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Introduction to Vertex Operator Algebras and Their Representations by James Lepowsky,Haisheng Li Pdf

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

Author : Karin Erdmann,Thorsten Holm
Publisher : Springer
Page : 304 pages
File Size : 49,5 Mb
Release : 2018-09-07
Category : Mathematics
ISBN : 9783319919980

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Algebras and Representation Theory by Karin Erdmann,Thorsten Holm Pdf

This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers. The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel's celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams. Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses.

Author : Alberto Facchini,Kent Fuller,Claus M Ringel,Catarina Santa-clara
Publisher : World Scientific
Page : 403 pages
File Size : 48,6 Mb
Release : 2006-02-20
Category : Mathematics
ISBN : 9789814478977

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Algebras, Rings And Their Representations - Proceedings Of The International Conference On Algebras, Modules And Rings by Alberto Facchini,Kent Fuller,Claus M Ringel,Catarina Santa-clara Pdf

Surveying the most influential developments in the field, this proceedings reviews the latest research on algebras and their representations, commutative and non-commutative rings, modules, conformal algebras, and torsion theories.The volume collects stimulating discussions from world-renowned names including Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith.

Author : R.W. Carter,J. Saxl
Publisher : Springer Science & Business Media
Page : 388 pages
File Size : 40,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401153089

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Algebraic Groups and their Representations by R.W. Carter,J. Saxl Pdf

This volume contains 19 articles written by speakers at the Advanced Study Institute on 'Modular representations and subgroup structure of al gebraic groups and related finite groups' held at the Isaac Newton Institute, Cambridge from 23rd June to 4th July 1997. We acknowledge with gratitude the financial support given by the NATO Science Committee to enable this ASI to take place. Generous financial support was also provided by the European Union. We are also pleased to acknowledge funds given by EPSRC to the Newton Institute which were used to support the meeting. It is a pleasure to thank the Director of the Isaac Newton Institute, Professor Keith Moffatt, and the staff of the Institute for their dedicated work which did so much to further the success of the meeting. The editors wish to thank Dr. Ross Lawther and Dr. Nick Inglis most warmly for their help in the production of this volume. Dr. Lawther in particular made an invaluable contribution in preparing the volume for submission to the publishers. Finally we wish to thank the distinguished speakers at the ASI who agreed to write articles for this volume based on their lectures at the meet ing. We hope that the volume will stimulate further significant advances in the theory of algebraic groups.

Author : H. Strade
Publisher : CRC Press
Page : 321 pages
File Size : 55,5 Mb
Release : 2020-08-12
Category : Mathematics
ISBN : 9781000146820

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Modular Lie Algebras and their Representations by H. Strade Pdf

This book presents an introduction to the structure and representation theory of modular Lie algebras over fields of positive characteristic. It introduces the beginner to the theory of modular Lie algebras and is meant to be a reference text for researchers.

Author : Seok-Jin Kang,Symposium on Lie Algebras,Myung-Hwan Kim,Insok Lee
Publisher : American Mathematical Soc.
Page : 242 pages
File Size : 41,7 Mb
Release : 1996
Category : Lie algebras
ISBN : 9780821805121

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Lie Algebras and Their Representations by Seok-Jin Kang,Symposium on Lie Algebras,Myung-Hwan Kim,Insok Lee Pdf

Over the past 30 years, exciting developments in diverse areas of the theory of Lie algebras and their representations have been observed. The symposium covered topics such as Lie algebras and combinatorics, crystal bases for quantum groups, quantum groups and solvable lattice models, and modular and infinite-dimensional Lie algebras. In this volume, readers will find several excellent expository articles and research papers containing many significant new results in this area.

Author : Robert N. Cahn
Publisher : Courier Corporation
Page : 180 pages
File Size : 51,8 Mb
Release : 2006-03-17
Category : Mathematics
ISBN : 9780486449999

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Semi-Simple Lie Algebras and Their Representations by Robert N. Cahn Pdf

Designed to acquaint students of particle physics already familiar with SU(2) and SU(3) with techniques applicable to all simple Lie algebras, this text is especially suited to the study of grand unification theories. Author Robert N. Cahn, who is affiliated with the Lawrence Berkeley National Laboratory in Berkeley, California, has provided a new preface for this edition. Subjects include the killing form, the structure of simple Lie algebras and their representations, simple roots and the Cartan matrix, the classical Lie algebras, and the exceptional Lie algebras. Additional topics include Casimir operators and Freudenthal's formula, the Weyl group, Weyl's dimension formula, reducing product representations, subalgebras, and branching rules. 1984 edition.

Author : A. Giambruno,Mikhail Zaicev,Michail V. Zaicev
Publisher : American Mathematical Soc.
Page : 370 pages
File Size : 51,9 Mb
Release : 2005
Category : PI-algebras
ISBN : 9780821838297

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Polynomial Identities and Asymptotic Methods by A. Giambruno,Mikhail Zaicev,Michail V. Zaicev Pdf

This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory. A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of co-dimensions of any PI-algebra has integral exponential growth - the PI-exponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent.

Author : Hermann Weyl
Publisher : Princeton University Press
Page : 335 pages
File Size : 50,6 Mb
Release : 1946
Category : Mathematics
ISBN : 9780691057569

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The Classical Groups by Hermann Weyl Pdf

The author discusses symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are important in understanding the group-theoretic structure of quantum mechanics.

Author : R Costa,Jr., Henrique Guzzo,A. Grichkov,L.A. Peresi
Publisher : CRC Press
Page : 492 pages
File Size : 51,9 Mb
Release : 2019-05-20
Category : Mathematics
ISBN : 9780429529993

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NonasSociative Algebra and Its Applications by R Costa,Jr., Henrique Guzzo,A. Grichkov,L.A. Peresi Pdf

A collection of lectures presented at the Fourth International Conference on Nonassociative Algebra and its Applications, held in Sao Paulo, Brazil. Topics in algebra theory include alternative, Bernstein, Jordan, lie, and Malcev algebras and superalgebras. The volume presents applications to population genetics theory, physics, and more.

Author : Leonid A. Bokut',Anatolij Ivanovič Mal'cev,A. I. Kostrikin,I︠U︡riĭ Leonidovich Ershov
Publisher : American Mathematical Soc.
Page : 712 pages
File Size : 44,7 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821851364

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Proceedings of the International Conference on Algebra Dedicated to the Memory of A. I. Mal$'$cev by Leonid A. Bokut',Anatolij Ivanovič Mal'cev,A. I. Kostrikin,I︠U︡riĭ Leonidovich Ershov Pdf