The Structure of Complex Lie Groups: 1st Edition (Paperback) book cover

The Structure of Complex Lie Groups

1st Edition

By Dong Hoon Lee

Chapman and Hall/CRC

232 pages | 10 B/W Illus.

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Description

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.

Table of Contents

COMPLEX LIE GROUPS

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

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT012000
MATHEMATICS / Geometry / General
MAT022000
MATHEMATICS / Number Theory
SCI040000
SCIENCE / Mathematical Physics