# Invariant Subsemigroups of Lie Groups

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*Karl-Hermann Neeb*

This work presents the first systematic treatment of invariant Lie semigroups. Because these semigroups provide interesting models for spacetimes in general relativity, this work will be useful to both mathematicians and physicists. It will also appeal to engineers interested in bi-invariant control systems on Lie groups. Neeb investigates closed invariant subsemigroups of Lie groups which are generated by one-parameter semigroups and the sets of infinitesimal generators of such semigroups—invariant convex cones in Lie algebras. In addition, a characterization of those finite-dimensional real Lie algebras containing such cones is obtained. The global part of the theory deals with globality problems (Lie's third theorem for semigroups), controllability problems, and the facial structure of Lie semigroups. Neeb also determines the structure of the universal compactification of an invariant Lie semigroup and shows that the lattice of idempotents is isomorphic to a lattice of faces of the cone dual to the cone of infinitesimal generators.

#### Table of Contents

# Table of Contents

## Invariant Subsemigroups of Lie Groups

- Table of Contents vii8 free
- Introduction 110 free
- I. Invariant Cones in K-modules 1524 free
- II. Lie Algebras with Cone Potential 2433
- III. Invariant Cones in Lie Algebras 4857
- IV. Faces of Lie Semigroups 7584
- V. Compactifications of Subsemigroups of Locally Compact Groups 92101
- VI. Invariant Subsemigroups of Lie Groups 97106
- VII. ControllabiKty of Invariant Wedges 116125
- VIII. Globality of Invariant Wedges 129138
- IX. Bohr Compactifications 146155
- X. The Unit Group of S[sup(b)] 167176
- XI. Faces and Idempotents 173182
- XII. Examples and Special Cases 182191
- References 190199