INVARIANT SUBSEMIGROUPS OF LIE-GROUPS

We study closed invariant subsemigroups S of Lie groups G which are Lie semigroups, i.e., topologically generated by one-parameter semigroups. Such a semigroup S is determined by its cone L(S) of infinitesimal generators, a dosed convex cone in the Lie algebra L(G) which is invariant under the adjoint action. First we investigate the structure of Lie algebras with invariant cones and give a characterization of those Lie algebras containing pointed and generating invariant cones. Then we study the global structure of invariant Lie semigroups, and how far Lie's third theorem remains true for invariant cones and Lie semigroups. Finally we describe the Bohr compactification S(b) of an invariant Lie semigroup. Most remarkably, the lattice of idempotents of S(b) is isomorphic to a certain lattice of faces of the cone dual to L(S).