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Invariant Subsemigroups of Lie Groups
Karl-Hermann Neeb

Memoirs of the American Mathematical Society
1993; 193 pp; softcover
Volume: 104
ISBN-10: 0-8218-2562-3
ISBN-13: 978-0-8218-2562-4
List Price: US$45
Individual Members: US$27
Institutional Members: US$36
Order Code: MEMO/104/499
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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.


Mathematicians interested in the geometry of cones, semigroups, and their compactifications.

Table of Contents

  • Introduction
  • Invariant cones in \(K\)-modules
  • Lie algebras with cone potential
  • Invariant cones in Lie algebras
  • Faces of Lie semigroups
  • Compactifications of subsemigroups of locally compact groups
  • Invariant subsemigroups of Lie groups
  • Controllability of invariant wedges
  • Globality of invariant wedges
  • Bohr compactifications
  • The unit group of \(S^\flat\)
  • Faces and idempotents
  • Examples and special cases
  • References
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