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English, Hardcover, 2009

Product Details

Author A.L.Onishchik

Press Science Press

Isbn 7030235045/9787030235046

Publication Date Jan 2009

Pages 235

Size 16K

Edition First

Cover Hardcover

Language English

Page material gelatine plate paper

Product Description:

The book by Gorbatsevich, Onishchik and Vinberg is the first volume in a subseries of the Encyclopaedia devoted to the theory of Lie groups and Lie algebras.

The first part of the book deals with the foundations of the theory based on the classical global approach of Chevalley followed by an exposition of the alternative approach via the universal enveloping algebra and the Campbell-Hausdorff formula. It also contains a survey of certain generalizations of Lie groups.

The second more advanced part treats the topic of Lie transformation groups covering e.g. properties of orbits and stabilizers, homogeneous fibre bundles, Frobenius duality, groups of automorphisms of geometric structures, Lie algebras of vector fields and the existence of slices. The work of the last decades including the most recent research results is covered.

The book contains numerous examples and describes connections with topology, differential geometry, analysis and applications. It is written for graduate students and researchers in mathematics and theoretical physics.

Content:

Introduction

Chapter 1. Basic Notions

1. Lie Groups, Subgroups and Homomorphisms

1.1 Definition of a Lie Group

1.2 Lie Subgroups

1.3 Homomorphisms of Lie Groups

1.4 Linear Representations of Lie Groups

1.5 Local Lie Groups

2. Actions of Lie Groups

2.1 Definition of an Action

2.2 Orbits and Stabilizers

2.3 Images and Kernels of Homomorphisms

2.4 Orbits of Compact Lie Groups

3. Coset Manifolds and Quotients of Lie Groups

3.1 Coset Manifolds

3.2 Lie Quotient Groups

3.3 The Transitive Action Theorem and the Epimorphism Theorem

3.4 The Pre-image of a Lie Group Under a Homomorphism

3.5 Semidirect Products of Lie Groups

4. Connectedness and Simply-connectedness of Lie Groups

4.1 Connected Components of a Lie Group

4.2 Investigation of Connectedness of the Classical Lie Groups

4.3 Covering Homomorphisms

4.4 The Universal Covering Lie Group

4.5 Investigation of Simply-connectedness of the Classical Lie Groups

Chapter 2. The Relation Between Lie Groups and Lie Algebras

1. The Lie Functor

1.1 The Tangent Algebra of a Lie Group

1.2 Vector Fields on a Lie Group

1.3 The Differential of a Homomorphism of Lie Groups

1.4 The Differential of an Action of a Lie Group

1.5 The Tangent Algebra of a Stabilizer

1.6 The Adjoint Representation

2. Integration of Homomorphisms of Lie Algebras

2.1 The Differential Equation of a Path in a Lie Group

2.2 The Uniqueness Theorem

2.3 Virtual Lie Subgroups

2.4 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra

2.5 Deformations of Paths in Lie Groups

2.6 The Existence Theorem

2.7 Abelian Lie Groups

3. The Exponential Map

3.1 One-Parameter Subgroups

3.2 Definition and Basic Properties of the Exponential Map

3.3 The Differential of the Exponential Map

3.4 The Exponential Map in the Full Linear Group

3.5 Cartan's Theorem

3.6 The Subgroup of Fixed Points of an Automorphism of a Lie Group

4. Automorphisms and Derivations

4.1 The Group of Automorphisms

4.2 The Algebra of Derivations

4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups

5. The Commutator Subgroup and the Radical

5.1 The Commutator Subgroup

5.2 The Maltsev Closure

5.3 The Structure of Virtual Lie Subgroups

5.4 Mutual Commutator Subgroups

5.5 Solvable Lie Groups

5.6 The Radical

5.7 Nilpotent Lie Groups

Chapter 3. The Universal Enveloping Algebra

1. The Simplest Properties of Universal Enveloping Algebras

1.1 Definition and Construction

1.2 The Poincare-Birkhoff-Witt Theorem

1.3 Symmetrization

1.4 The Center of the Universal Enveloping Algebra

1.5 The Skew-Field of Fractions of the Universal Enveloping Algebra

2. Bialgebras Associated with Lie Algebras and Lie Groups

2.1 Bialgebras

2.2 Right Invariant Differential Operators on a Lie Group

2.3 Bialgebras Associated with a Lie Group

3. The Campbell-Hausdorff Formula

3.1 Free Lie Algebras

3.2 The Campbell-Hausdorff Series

3.3 Convergence of the Campbell-Hausdorff Series

Chapter 4. Generalizations of Lie Groups

1. Lie Groups over Complete Valued Fields

1.1 Valued Fields

1.2 Basic Definitions and Examples

1.3 Actions of Lie Groups

1.4 Standard Lie Groups over a Non-archimedean Field

1.5 Tangent Algebras of Lie Groups

2. Formal Groups

2.1 Definition and Simplest Properties

2.2 The Tangent Algebra of a Formal Group

2.3 The Bialgebra Associated with a Formal Group

3. Infinite-Dimensional Lie Groups

3.1 Banach Lie Groups

3.2 The Correspondence Between Banach Lie Groups and Banach Lie Algebras

3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds

3.4 Lie-Frechet Groups

3.5 ILB- and ILH-Lie Groups

4. Lie Groups and Topological Groups

4.1 Continuous Homomorphisms of Lie Groups

4.2 Hilbert's 5-th Problem

5. Analytic Loops

5.1 Basic Definitions and Examples

5.2 The Tangent Algebra of an Analytic Loop

5.3 The Tangent Algebra of a Diassociative Loop

5.4 The Tangent Algebra of a Bol Loop

References

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