The Structure of Complex Lie Groups

The Structure of Complex Lie Groups

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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 more

Product details

  • Paperback | 232 pages
  • 155.7 x 233.7 x 14mm | 344.74g
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 10 black & white illustrations
  • 1584882611
  • 9781584882619

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 Groupsshow more