The Structure of Complex Lie Groups, Vol. 429 Book

	Preface
1	Complex Lie Groups	1
1.1	Analytic Manifolds	1
1.2	Complex Lie Groups	12
1.3	Examples of Complex Lie Groups	24
1.4	The Automorphism Group	35
1.5	Universal Complexification	39
2	Representative Functions of Lie Groups	49
2.1	Basic Definitions	49
2.2	Algebras of Representative Functions	55
2.3	Proper Automorphism Groups	60
2.4	Analytic Representative Functions	64
2.5	Universal Algebraic Hull	68
2.6	Relative Algebras	73
2.7	Unipotent Hull	77
3	Extensions of Representations	81
3.1	Some Examples	81
3.2	Decomposition Theorem	84
3.3	Main Lemma	88
3.4	Extensions of Representations	91
3.5	Unipotent Analytic Groups	94
3.6	Application to Solvable Groups	96
4	The Structure of Complex Lie Groups	99
4.1	Abelian Complex Analytic Groups	99
4.2	Decomposition of the Adjoint Group	104
4.3	Semisimple Complex Analytic Groups	112
4.4	Reductive Complex Analytic Groups	116
4.5	Compact Subgroups of Reductive Groups	119
4.6	Representation Radical	125
4.7	Faithfully Representable Groups	129
4.8	Conjugacy of Reductive Subgroups	132
4.9	Unipotent Hull of Complex Lie Groups	134
5	Algebraic Subgroups in Lie Groups	137
5.1	Affine Algebraic Structure in Lie Groups	137
5.2	Extension Lemma	139
5.3	Affine Algebraic Structure on Reductive Groups	142
5.4	The Maximum Algebraic Subgroup	148
5.5	Further Properties of Reductive Groups	154
6	Observability in Complex Lie Groups	157
6.1	Pro-affine Groups and Observability	157
6.2	Observability in Algebraic Groups	165
6.3	Extension of Representative Functions	171
6.4	Structure of Observable Subgroups	174
A	Elementary Theory of Lie Algebras	183
B	Pro-affine Algebraic Groups	197
	Bibliography	211
	List of Notations	215
	Index	217