Lie Semigroups and Their ApplicationsSubsemigroups of finite-dimensional Lie groups that are generated by one-parameter semigroups are the subject of this book. It covers basic Lie theory for such semigroups and some closely related topics. These include ordered homogeneous manifolds, where the order is defined by a field of cones, invariant cones in Lie algebras and associated Ol'shanskii semigroups. Applications to representation theory, symplectic geometry and Hardy spaces are also given. The book is written as an efficient guide for those interested in subsemigroups of Lie groups and their applications in various fields of mathematics (see the User's guide at the end of the Introduction). Since it is essentially self-contained and leads directly to the core of the theory, the first part of the book can also serve as an introduction to the subject. The reader is merely expected to be familiar with the basic theory of Lie groups and Lie algebras. |
Contents
Lie semigroups and their tangent wedges | 1 |
Geometry and topology of Lie semigroups | 3 |
Invariant Cones and Olshanski semigroups | 7 |
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abelian algebra g algint analytic assume Aut(C Cartan algebra Cartan subalgebra closed submonoid closed subsemigroup compact subset compactly embedded compr(C conal curve cone field connected Lie group contains convex Corollary define denote dense element exists exposed face Fe(W finite dimensional flag manifold following assertions hold function f G-orbit G₁ global in G globally orderable Haar measure Hence Hermitean highest weight holomorphic homogeneous spaces implies int(C int(S invariant cone isomorphic Lemma Let G Lie algebra Lie group G Lie semigroup Lie wedge Lorentzian mapping maximal subsemigroup Mon(S monotone functions morphism neighborhood non-compact ordered homogeneous spaces pointed generating invariant Proof Proposition reach(S representation S-monotone S₁ Section sequence shows simply connected strictly positive subalgebra submonoid subsemigroup subsemigroup of G subspace Suppose tangent wedge Theorem topological group universal covering universal covering group vector space W₁