Lie Groups

Preface
	Acknowledgement
Contents
1 Introduction
	1.1 Exercises
Part I Topological Groups
	Overview
	2 Topological Groups
		2.1 Introduction
		2.2 Neighborhoods of Identity
		2.3 Metrizable Groups
		2.4 Homomorphisms
		2.5 Subgroups
		2.6 Group Actions
			2.6.1 Algebraic Description
			2.6.2 Continuous Actions
		2.7 Quotient Spaces
			2.7.1 Quotient Groups
			2.7.2 Compact and Connected Groups
		2.8 Homeomorphism G/Gx→G·x
		2.9 Examples
		2.10 Exercises
	3 Haar Measure
		3.1 Introduction
		3.2 Construction of Haar Measure
		3.3 Uniqueness
		3.4 Modular Function
		3.5 Exercises
	4 Representations of Compact Groups
		4.1 Representations
		4.2 Schur Orthogonality Relations
		4.3 Regular Representations
		4.4 Peter–Weyl Theorem
		4.5 Exercises
Part II Lie Groups and Algebras
	Overview
	5 Lie Groups and Lie Algebras
		5.1 Definition
		5.2 Lie Algebra of a Lie Group
			5.2.1 Invariant Vector Fields
		5.3 Exponential Map
		5.4 Homomorphisms
			5.4.1 Representations
			5.4.2 Adjoint Representations
		5.5 Ordinary Differential Equations
		5.6 Haar Measure
		5.7 Exercises
	6 Lie Subgroups
		6.1 Definition and Examples
		6.2 Lie Subalgebras and Lie Subgroups
		6.3 Ideals and Normal Subgroups
		6.4 Limits of Products of Exponentials
		6.5 Closed Subgroups
		6.6 Path Connected Subgroups
		6.7 Manifold Structure on G/H, H Closed
		6.8 Exercises
	7 Homomorphisms and Coverings
		7.1 Homomorphisms
			7.1.1 Immersions and Submersions
			7.1.2 Graphs and Differentiability
		7.2 Extensions of Homomorphisms
		7.3 Universal Covering
		7.4 Appendix: Covering Spaces (Overview)
		7.5 Exercises
	8 Series Expansions
		8.1 The Differential of the Exponential Map
		8.2 The Baker–Campbell–Hausdorff Series
		8.3 Analytic-Manifold Structure
		8.4 Exercises
Part III Lie Algebras and Simply Connected Groups
	Overview
	9 The Affine Group and Semi-Direct Products
		9.1 Automorphisms of Lie Groups
		9.2 The Affine Group
		9.3 Semi-Direct Products
		9.4 Derived Groups and Lower Central Series
		9.5 Exercises
	10 Solvable and Nilpotent Groups
		10.1 Solvable Groups
		10.2 Nilpotent Groups
		10.3 Exercises
	11 Compact Groups
		11.1 Compact Lie Algebras
		11.2 Finite Fundamental Group
			11.2.1 Extension Theorem
		11.3 Compact and Complex Lie Algebras
			11.3.1 Weyl Unitary Trick
			11.3.2 Dynkin Diagrams
			11.3.3 Cartan Subalgebras and Regular Elements
		11.4 Maximal Tori
		11.5 Center and Roots
		11.6 Riemannian Geometry
		11.7 Exercises
	12 Noncompact Semi-Simple Groups
		12.1 Cartan Decompositions
			12.1.1 Cartan Decomposition of a Lie Algebra
			12.1.2 Global Cartan Decomposition
		12.2 Iwasawa Decompositions
			12.2.1 Iwasawa Decomposition of a Lie Algebra
			12.2.2 Global Iwasawa Decomposition
		12.3 Classification
		12.4 Exercises
Part IV Transformation Groups
	Overview
	13 Lie Group Actions
		13.1 Group Actions
			13.1.1 Orbits
		13.2 Lie–Palais Theorem
			13.2.1 Families of Vector Fields
		13.3 Bundles
			13.3.1 Principal Bundles
			13.3.2 Associated Bundles
		13.4 Homogeneous Spaces and Bundles
		13.5 Exercises
	14 Invariant Geometry
		14.1 Complex Manifolds
			14.1.1 Complex Lie Groups
		14.2 Differential Forms and de Rham Cohomology
		14.3 Riemannian Manifolds
		14.4 Symplectic Manifolds
			14.4.1 Coadjoint Representation
			14.4.2 Moment Map
		14.5 Exercises
Part V Appendices
A Vector Fields and Lie Brackets
	A.1 Exercises
B Integrability of Distributions
	B.1 Immersions and Submanifolds
	B.2 Characteristic Distributions
	B.3 Maximal Integral Manifolds
	B.4 Adapted Charts
	B.5 Integral Manifolds Are Quasi-Regular
	B.6 Exercises
References
Index