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Identities of Algebras and their Representations
Yu. P. Razmyslov
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Translations of Mathematical Monographs
1994; 318 pp; hardcover
Volume: 138
ISBN-10: 0-8218-4608-6
ISBN-13: 978-0-8218-4608-7
List Price: US$127
Member Price: US$101.60
Order Code: MMONO/138
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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 \(\alpha\)-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

"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

  • Preliminary results
  • Characters and \(\alpha\)-functions on 2 -words and varieties of representations of Lie algebras distinguished by them
  • \(\alpha\)-functions related to the killing form and to irreducible representations of semisimple Lie algebras. Central polynomials of irreducible representations of reductive Lie algebras
  • \(\alpha\)-functions related to full matrix algebras. Trace identities and central polynomials of full matrix algebras \(M_n\) and matrix superalgebras \(M_{n,k}\)
  • The \(\alpha\)-function related to representations of the simple three-dimensional Lie algebra \(\mathfrak g\) and its applications to varieties of groups and associative algebras
  • Varieties generated by Lie algebras of Cartan type
  • Algebraic supplements
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