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Invariant Subsemigroups of Lie Groups

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Electronic ISBN: 978-1-4704-0076-7
Product Code: MEMO/104/499.E
List Price: $45.00 MAA Member Price:$40.50
AMS Member Price: $27.00 Click above image for expanded view Invariant Subsemigroups of Lie Groups Available Formats:  Electronic ISBN: 978-1-4704-0076-7 Product Code: MEMO/104/499.E  List Price:$45.00 MAA Member Price: $40.50 AMS Member Price:$27.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1041993; 193 pp
MSC: Primary 22; 43;

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.

• Chapters
• Introduction
• I. Invariant cones in $K$-modules
• II. Lie algebras with cone potential
• III. Invariant cones in Lie algebras
• IV. Faces of Lie semigroups
• V. Compactifications of subsemigroups of locally compact groups
• VI. Invariant subsemigroups of Lie groups
• VII. Controllability of invariant wedges
• VIII. Globality of invariant wedges
• IX. Bohr compactifications
• X. The unit group of $S^\flat$
• XI. Faces and idempotents
• XII. Examples and special cases
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Volume: 1041993; 193 pp
MSC: Primary 22; 43;

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.

• Chapters
• Introduction
• I. Invariant cones in $K$-modules
• II. Lie algebras with cone potential
• III. Invariant cones in Lie algebras
• IV. Faces of Lie semigroups
• V. Compactifications of subsemigroups of locally compact groups
• VI. Invariant subsemigroups of Lie groups
• VII. Controllability of invariant wedges
• VIII. Globality of invariant wedges
• IX. Bohr compactifications
• X. The unit group of $S^\flat$
• XI. Faces and idempotents
• XII. Examples and special cases
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