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Lie Semigroups and Their Applications : Lecture Notes in Mathematics - Joachim Hilgert

Lie Semigroups and Their Applications

Lecture Notes in Mathematics


Published: 29th September 1993
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Subsemigroups of finite-dimensional Lie groups that aregenerated by one-parameter semigroups are the subject ofthis book. It covers basic Lie theory for such semigroupsand some closely related topics. These include orderedhomogeneous manifolds, where the order is defined by a fieldof cones, invariant cones in Lie algebras and associatedOl'shanskii semigroups. Applications to representationtheory, symplectic geometry and Hardy spaces are also given.The book is written as an efficient guide for thoseinterested in subsemigroups of Lie groups and theirapplications in various fields of mathematics (see theUser's guide at the end of the Introduction). Since it isessentially self-contained and leads directly to the core ofthe theory, the first part of the book can also serve as anintroduction to the subject.The reader is merely expected to be familiar with the basictheory of Lie groups and Lie algebras.

Lie semigroups and their tangent wedges
Geometry of wedgesp. 1
Wedges in K-modulesp. 9
The characteristic function of a conep. 11
Lie wedges and Lie semigroupsp. 19
Functorial relations between Lie semigroups and Lie wedgesp. 25
Globality of Lie wedgesp. 28
Monotone functions and semigroupsp. 29
Smooth and analytic monotone functions on a Lie groupp. 32
W-positive functions and globalityp. 38
Globality criteriap. 42
Semigroups in the Heisenberg groupp. 48
The groups SI(2) and PSI(2, R)p. 49
The hyperboloid and its order structurep. 56
The Olshanskii semigroup in Sl(2,C)p. 59
Affine compression semigroupsp. 61
The euclidean compression and contraction semigroupsp. 63
Godel's cosmological model and the universal covering of Sl(2, R)p. 65
The causal action of SU(n, n) on U(n)p. 68
Almost abelian groupsp. 73
The whirlpot and the parking rampp. 73
The oscillator groupp. 76
Geometry and topology of Lie semigroups
Faces of Lie semigroupsp. 81
The interior of Lie semigroupsp. 86
Non generating Lie semigroups with interior pointsp. 88
The universal covering semigroup [actual symbol not reproducible]p. 90
The free group on Sp. 101
Groups with directed ordersp. 107
Ordered homogeneous spaces
Chains in metric pospacesp. 114
Invariant cone fields on homogeneous spacesp. 121
Globality of cone fieldsp. 126
Chains and conal curvesp. 130
Covering spaces and globalityp. 135
Regular ordered homogeneous spacesp. 138
Extremal curvesp. 140
Applications of ordered spaces to Lie semigroups
Consequences of the Globality Theoremp. 148
Consequences of the Covering Theoremp. 149
Conal curves and reachability in semigroupsp. 153
Applications to faces of Lie semigroupsp. 156
Monotone curves in Lie semigroupsp. 159
Maximal semigroups in groups with cocompact radical
Hyperplane subalgebras of Lie algebrasp. 162
Elementary facts about maximal semigroupsp. 164
Abelian and almost abelian groupsp. 166
Nilpotent groupsp. 167
Reduction lemmasp. 169
Characterization of maximal subsemigroupsp. 171
Applications to reachability questionsp. 173
Invariant Cones and Ol'shanskii semigroups
Compactly embedded Cartan algebrasp. 177
Invariant cones in Lie algebrasp. 184
Lawson's Theorem on Olshanskii semigroupsp. 194
Compression semigroups
Invariant control setsp. 203
Moment maps and projective spacesp. 209
Pseudo-unitary representations and orbits on flag manifoldsp. 218
Compression semigroups of open G-orbitsp. 232
Contraction semigroups for indefinite formsp. 246
Maximality of complex Ol'shanskii semigroupsp. 250
Representation theory
Involutive semigroupsp. 254
Holomorphic representations of half planesp. 257
Invariant cones and unitary representationsp. 262
Holomorphic discrete series representationsp. 269
Hardy spacesp. 276
Howe's oscillator semigroupp. 288
The Luscher-Mack Theoremp. 291
The theory for Sl(2)
Referencesp. 303
List of Symbolsp. 311
Indexp. 313
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540569541
ISBN-10: 3540569545
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 316
Published: 29th September 1993
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.75
Weight (kg): 0.47