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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