The Structure of Complex Lie Groups

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.

COMPLEX LIE GROUPSAlmost Complex StructureComplex Lie GroupsExamples of Complex Lie GroupsAutomorphism Groups and Semidirect ProductsUniversal Complexification of Real Lie GroupsREPRESENTATIVE FUNCTIONS OF LIE GROUPSBasic Definitions of RepresentationsRepresentative Functions and Proper AutomorphismsAnalytic Representative FunctionsUniversal Algebraic HullRelative Algebras of Representative FunctionsUnipotent HullEXTENSION OF REPRESENTATIONSSome ExamplesDecomposition of R(G)Extension LemmasExtensions of RepresentationsApplication of Extension TheoremTHE STRUCTURE OF COMPLEX LIE GROUPSAbelian Complex Analytic GroupsSemisimple Complex Analytic GroupsReductive Complex Analytic GroupsMaximal Compact Subgroups and ReductivityRepresentation Radical of Analytic GroupsFaithfully Representable GroupsConjugacy of Reductive SubgroupsUnipotent Hull of Faithfully Representable GroupsALGEBRAIC SUBGROUPS OF COMPLEX LIE GROUPSAlgebraic Subgroups of Analytic GroupsExtension of Representations and Representative FunctionsAlgebraic Group Structure of Reductive SubgroupsThe Maximal Algebraic SubgroupFurther Properties of Reductive GroupsOBSERVABILITY IN COMPLEX ANALYTIC GROUPSPro-Affine Algebraic Groups and ObservabilityAffine Algebraic Groups and ObservabilityAlgebraic Hull of Observable Analytic SubgroupsExtension of Analytic Representative FunctionsStructure of Observable Subgroups of Complex Lie GroupsAPPENDIX 1: Elementary Theory of Lie AlgebrasAPPENDIX 2: Pro-Affine Algebraic Groups