Identities of Algebras and their Representations
Identities of Algebras and their Representations

Author: I︠U︡riĭ Pitrimovich Razmyslov

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 468

ISBN-13: 9780821846087

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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.

Polynomial Identities in Algebras
Polynomial Identities in Algebras

Author: Onofrio Mario Di Vincenzo

Publisher: Springer Nature

Published: 2021-03-22

Total Pages: 421

ISBN-13: 3030631117

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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.

Algebras and Representation Theory
Algebras and Representation Theory

Author: Karin Erdmann

Publisher: Springer

Published: 2018-09-07

Total Pages: 304

ISBN-13: 3319919989

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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.

Three Papers on Algebras and Their Representations
Three Papers on Algebras and Their Representations

Author: V. N. Gerasimov

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 226

ISBN-13: 9780821875032

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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.

Introduction to Vertex Operator Algebras and Their Representations
Introduction to Vertex Operator Algebras and Their Representations

Author: James Lepowsky

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 330

ISBN-13: 0817681868

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* 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.

Rings with Polynomial Identities and Finite Dimensional Representations of Algebras
Rings with Polynomial Identities and Finite Dimensional Representations of Algebras

Author: Eli Aljadeff

Publisher: American Mathematical Soc.

Published: 2020-12-14

Total Pages: 630

ISBN-13: 1470451743

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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.

Introduction to Representation Theory
Introduction to Representation Theory

Author: Pavel I. Etingof

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 240

ISBN-13: 0821853511

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Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.

Algebraic Groups and their Representations
Algebraic Groups and their Representations

Author: R.W. Carter

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 388

ISBN-13: 9401153086

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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.