Modular Lie Algebras and their Representations

H. Strade

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780824775940
• eBook ISBN: 9781000146820
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

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Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Contents
• Chapter 1 Classical Theory
• 1.1 Basic Definitions of Lie Algebras and Representations
• 1.2 Some Basic Module Operations
• 1.3 Nilpotent Lie Algebras; Engel's Theorem
• 1.4 Primary Decomposition; Cartan Subalgebras
• 1.5 Solvable Lie Algebras; Lie's Theorem
• 1.6 Examples of Low Dimension
• 1.7 The Solvable Radical and Cartan's Criterion for Solvability
• 1.8 The Universal Enveloping Algebra
• 1.9 Filtrations
• References
• Chapter 2 Basic Theory of Restrictable Lie Algebras
• 2.1 p-Mappings and Restricted Lie Algebras
• 2.2 Existence of p-Mappings; Restrictable Lie Algebras
• 2.3 Some Properties of the p-Mapping
• 2.4 Tori and Cartan Decomposition
• 2.5 Restricted Enveloping Algebras and Universal p-Envelopes
• References
• Chapter 3 Filtered and Graded Lie Algebras
• 3.1 Filtered Lie Algebras
• 3.2 Graded Lie Algebras
• 3.3 The Interrelation Between Filtrations and Gradations; Criteria for Simplicity
• 3.4 Subsidiary Results on Filtrations and Gradations
• 3.5 Realizations as Derivation Algebras
• References
• Chapter 4 Graded Lie Algebras of Cartan Type
• 4.1 The Construction Processes
• 4.2 The Generali zed Jacobson-Witt Algebra W(n;m.)
• 4.3 The Special Algebra S(n;m.)
• 4.4 The Hamiltonian Algebra H(2r;m.)
• 4.5 The Contact Algebra K(2r + l; m.)
• 4.6 Associative Forms of Graded Cartan-Type Lie Algebras
• 4.7 Generators of Restricted Cartan-Type Lie Algebras
• 4.8 The Derivation Algebras of the Restricted Cartan-Type Lie Algebras
• References
• Chapter 5 Representations of Modular Lie Algebras
• 5.1 The Center of the Universal Enveloping Algebra
• 5.2 Irreducible Representations
• 5.3 Reduced Enveloping Algebras
• 5.4 Frobenius Algebras
• 5.5 Lie Algebras with Completely Reducible Representations
• 5.6 Induced Representations
• 5.7 A Criterion for Irreducibility
• 5.8 Representations of Solvable Lie Algebras
• 5.9 Examples
• References
• Chapter 6 The Ideal Structure of The Universal Enveloping Algebra
• 6.1 The Prime Spectrum of a Ring
• 6.2 Noetherian Rings
• 6.3 Properties of the Mapping Spec(A) . Spec(R)
• 6.4 The Going-Down Property
• 6.5 The Prime Spectrum of U(L)
• 6.6 The Maximal Dimension of Irreducible Modules
• 6.7 Examples
• References
• Notation
• Bibliography
• Index