Lie Groups and Lie Algebras I: Foundations of Lie Theory Lie Transformation Groups (Paperback, Softcover Repri)

I.Foundations of Lie Theory.- 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.- 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.- 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.- 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.