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Modular Lie algebras and their representations. (English) Zbl 0648.17003
This book provides a readable modern account of modular Lie algebras, an area in which major developments have taken place over the past decade. One finds the material needed to pursue the study of the Block-Wilson classification for simple restricted Lie algebras but this is not included in the text.
The first chapter is an introduction to classical theory which both introduces basic concepts and allows for comparison as the modular theory is developed. The next three chapters provide the background needed for the classification of simple restricted Lie algebras. Chapter two collects basic facts, emphasizes the concept of an algebra being restrictable, describes Cartan subalgebras by maximal tori and introduces the Jordan-Chevalley-Seligman decomposition. The third chapter deals with filtrations and gradations which have been important tools in the classification problem. Chapter four introduces the simple Lie algebras of Cartan type, those with degenerate Killing from. These algebras are studied via their generators, their forms and their derivations.
The emphasis in the final two chapters is toward representation theory which leads to the universal envelope, associative techniques and the prime spectrum. Much information on irreducible representations is obtained under various assumptions.
In summary, this excellent book is a valuable
Reviewer: E.L.Stitzinger

17B50 Modular Lie (super)algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras
17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)