**Translations of Mathematical Monographs**

1994;
318 pp;
Hardcover

MSC: Primary 17; 16;

Print ISBN: 978-0-8218-4608-7

Product Code: MMONO/138

List Price: $127.00

AMS Member Price: $101.60

MAA Member Price: $114.30

**Electronic ISBN: 978-1-4704-4546-1
Product Code: MMONO/138.E**

List Price: $127.00

AMS Member Price: $101.60

MAA Member Price: $114.30

# Identities of Algebras and their Representations

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*Yu. P. Razmyslov*

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 \(\alpha\)-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 \(\alpha\)-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.

#### Readership

Research mathematicians.

#### Reviews & Endorsements

Yuri P. Razmyslov has been one of the most active in this work over the last 20 years, and in this book he gives a connected account of his methods … highly original work … a major work that should be in every mathematical library.

-- Bulletin of the London Mathematical Society

The author … is one of the Russian mathematicians who has done much to give modern ring and group theory a contemporary face … the reader will be rewarded with the pleasure of touching from the same point of view nice and important results of different fields of mathematics.

-- Mathematical Reviews

Provides an excellent opportunity for specialists in ring theory to become familiar with … significant advances in the theory of algebras satisfying polynomial identities.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Identities of Algebras and their Representations

- Cover Cover11
- Title page v6
- Contents vii8
- Preface xi12
- Chapter I. Preliminary results 116
- Chapter II. Characters and ๐ผ-functions on 2-words and varieties of representations of Lie algebras distinguished by them 3752
- Chapter III. ๐ผ-functions related to the killing form and to irreducible representations of semisimple Lie algebras. Central polynomials of irreducible representations of reductive Lie algebras 7186
- Chapter IV. ๐ผ-functions related to full matrix algebras. Trace identities and central polynomials of full matrix algebras ๐_{๐} and matrix superalgebras ๐_{๐,๐} 103118
- Chapter V. The ๐ผ-function related to representations of the simple three-dimensional Lie algebra ๐ค and its applications to varieties of groups and associative algebras 161176
- Chapter VI. Varieties generated by Lie algebras of Cartan type 211226
- Chapter VII. Algebraic supplements 265280
- Historical survey 305320
- References 313328
- Back Cover Back Cover1335