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Invariant subsemigroups of Lie groups. (English) Zbl 0786.22001
Mem. Am. Math. Soc. 499, 193 p. (1993).
The purpose of this monograph is to present a detailed study of invariant Lie subsemigroups \(S\) in a Lie group \(G\) (\(S\) is said to be invariant if \(S\) is invariant under all inner automorphisms and is a Lie subsemigroup if \(S\) is topologically generated by its one-parameter subsemigroups). A Lie subsemigroup is densely determined by its cone \(L(S)\) of infinitesimal generators which must be a closed convex cone in the Lie algebra \(L(G)\) which is invariant under the adjoint action. The first part of the monograph contains a detailed study and characterization of those Lie algebras containing an invariant pointed generating cone. For simple Lie algebras, these are the Hermitian algebras, and certain of their properties (compact subalgebras having a center, compact Cartan subalgebras) play a crucial role for the general case. The author also extends the earlier work of J. Hilgert and K. H. Hofmann [Adv. Math. 75, 168-201 (1989; Zbl 0685.22004)] on characterizing an invariant cone in terms of its intersection with a compactly embedded Cartan algebra.
The author next considers the problem of deciding for a given invariant cone, whether there is a Lie subsemigroup in the corresponding simply connected Lie group with the cone as tangent cone (a semigroup variant of Lie’s third fundamental theorem). Conditions are derived for this to hold, but under other conditions it is shown the invariant cone is controllable, i.e., generates the entire Lie group.
Finally, the author considers the Bohr compactification of an invariant Lie semigroup \(S\). It is shown that the lattice of idempotents is isomorphic to a certain lattice of faces of the cone dual to the tangent cone \(L(S)\).

MSC:
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22A15 Structure of topological semigroups
22E15 General properties and structure of real Lie groups
22E60 Lie algebras of Lie groups
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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