Complex Lie groups have often been used as auxiliaries in the study of real Lie groups in areas such as differential geometry and representation theory. To date, however, no book has fully explored and developed their structural aspects.
The Structure of Complex Lie Groups addresses this need. Self-contained, it begins with general concepts introduced via an almost complex structure on a real Lie group. It then moves to the theory of representative functions of Lie groups- used as a primary tool in subsequent chapters-and discusses the extension problem of representations that is essential for studying the structure of complex Lie groups. This is followed by a discourse on complex analytic groups that carry the structure of affine algebraic groups compatible with their analytic group structure. The author then uses the results of his earlier discussions to determine the observability of subgroups of complex Lie groups.
The differences between complex algebraic groups and complex Lie groups are sometimes subtle and it can be difficult to know which aspects of algebraic group theory apply and which must be modified. The Structure of Complex Lie Groups helps clarify those distinctions. Clearly written and well organized, this unique work presents material not found in other books on Lie groups and serves as an outstanding complement to them.
Almost Complex Structure
Complex Lie Groups
Examples of Complex Lie Groups
Automorphism Groups and Semidirect Products
Universal Complexification of Real Lie Groups
REPRESENTATIVE FUNCTIONS OF LIE GROUPS
Basic Definitions of Representations
Representative Functions and Proper Automorphisms
Analytic Representative Functions
Universal Algebraic Hull
Relative Algebras of Representative Functions
Unipotent Hull
EXTENSION OF REPRESENTATIONS
Some Examples
Decomposition of R(G)
Extension Lemmas
Extensions of Representations
Application of Extension Theorem
THE STRUCTURE OF COMPLEX LIE GROUPS
Abelian Complex Analytic Groups
Semisimple Complex Analytic Groups
Reductive Complex Analytic Groups
Maximal Compact Subgroups and Reductivity
Representation Radical of Analytic Groups
Faithfully Representable Groups
Conjugacy of Reductive Subgroups
Unipotent Hull of Faithfully Representable Groups
ALGEBRAIC SUBGROUPS OF COMPLEX LIE GROUPS
Algebraic Subgroups of Analytic Groups
Extension of Representations and Representative Functions
Algebraic Group Structure of Reductive Subgroups
The Maximal Algebraic Subgroup
Further Properties of Reductive Groups
OBSERVABILITY IN COMPLEX ANALYTIC GROUPS
Pro-Affine Algebraic Groups and Observability
Affine Algebraic Groups and Observability
Algebraic Hull of Observable Analytic Subgroups
Extension of Analytic Representative Functions
Structure of Observable Subgroups of Complex Lie Groups
APPENDIX 1: Elementary Theory of Lie Algebras
APPENDIX 2: Pro-Affine Algebraic Groups
Biography
Dong Hoon Lee
