Lie Groups: An Introduction Through Linear Groups Book

1	The exponential map	1
1.1	Vector fields and one-parameter groups of linear transformation	1
1.2	Ad, ad, and d exp	12
1.3	The Campbell-Baker-Hausdorff series	22
2	Lie theory	30
2.1	Linear groups: definitions and examples	30
2.2	The Lie algebra of a linear group	44
2.3	Coordinates on a linear group	53
2.4	Connectedness	61
2.5	The Lie correspondence	66
2.6	Homomorphisms and coverings of linear groups	78
2.7	Closed subgroups	87
3	The classical groups	91
3.1	The classical groups: definitions, connectedness	91
3.2	Cartan subgroups	107
3.3	Roots, weights, reflections	115
3.4	Fundamental groups of the classical groups	121
4	Manifolds, homogeneous spaces, Lie groups	132
4.1	Manifolds	132
4.2	Homogeneous spaces	143
4.3	Lie groups	152
5	Integration	165
5.1	Integration on manifolds	165
5.2	Integration on linear groups and their homogeneous spaces	171
5.3	Weyl's integration formula for U(n)	179
6	Representations	189
6.1	Representations: definitions	189
6.2	Schur's lemma, Peter-Weyl theorem	197
6.3	Characters	205
6.4	Weyl's character formula for U(n)	212
6.5	Representations of Lie algebras	223
6.6	The Borel-Weil theorem for GL(n,C)	232
6.7	Representations of the classical groups	237
App	Analytic Functions and Inverse Function Theorem	250
	References	258
	Index	261