Subsemigroups Of Finite-dimensional Lie Groups That Are Generated By One-parameter Semigroups Are The Subject Of This Book. It Covers Basic Lie Theory For Such Semigroups And Some Closely Related Topics. These Include Ordered Homogeneous Manifolds, Where The Order Is Defined By A Field Of Cones, Invariant Cones In Lie Algebras And Associated Ol'shanskii Semigroups. Applications To Representation Theory, Symplectic Geometry And Hardy Spaces Are Also Given. The Book Is Written As An Efficient Guide For Those Interested In Subsemigroups Of Lie Groups And Their Applications In Various Fields Of Mathematics (see The User's Guide At The End Of The Introduction). Since It Is Essentially Self-contained And Leads Directly To The Core Of The Theory, The First Part Of The Book Can Also Serve As An Introduction To The Subject. The Reader Is Merely Expected To Be Familiar With The Basic Theory Of Lie Groups And Lie Algebras. 1. Lie Semigroups And Their Tangent Wedges -- 1.1. Geometry Of Wedges -- 1.2. Wedges In K-modules -- 1.3. The Characteristic Function Of A Cone -- 1.4. Lie Wedges And Lie Semigroups -- 1.5. Functorial Relations Between Lie Semigroups And Lie Wedges -- 1.6. Globality Of Lie Wedges -- 1.7. Monotone Functions And Semigroups -- 1.8. Smooth And Analytic Monotone Functions On A Lie Group -- 1.9. W-positive Functions And Globality -- 1.10. Globality Criteria -- 2. Examples -- 2.1. Semigroups In The Heisenberg Group -- 2.2. The Groups Si(2) And Psi(2, R) -- 2.3. The Hyperboloid And Its Order Structure -- 2.4. The Olshanskii Semigroup In Sl(2,c) -- 2.5. Affine Compression Semigroups -- 2.6. The Euclidean Compression And Contraction Semigroups -- 2.7. Godel's Cosmological Model And The Universal Covering Of Sl(2, R) -- 2.8. The Causal Action Of Su(n, N) On U(n) -- 2.9. Almost Abelian Groups -- 2.10. The Whirlpot And The Parking Ramp -- 2.11. The Oscillator Group --^ 3. Geometry And Topology Of Lie Semigroups -- 3.1. Faces Of Lie Semigroups -- 3.2. The Interior Of Lie Semigroups -- 3.3. Non Generating Lie Semigroups With Interior Points -- 3.4. The Universal Covering Semigroup [actual Symbol Not Reproducible] -- 3.5. The Free Group On S -- 3.6. Groups With Directed Orders -- 4. Ordered Homogeneous Spaces -- 4.1. Chains In Metric Pospaces -- 4.2. Invariant Cone Fields On Homogeneous Spaces -- 4.3. Globality Of Cone Fields -- 4.4. Chains And Conal Curves -- 4.5. Covering Spaces And Globality -- 4.6. Regular Ordered Homogeneous Spaces -- 4.7. Extremal Curves -- 5. Applications Of Ordered Spaces To Lie Semigroups -- 5.1. Consequences Of The Globality Theorem -- 5.2. Consequences Of The Covering Theorem -- 5.3. Conal Curves And Reachability In Semigroups -- 5.4. Applications To Faces Of Lie Semigroups -- 5.5. Monotone Curves In Lie Semigroups -- 6. Maximal Semigroups In Groups With Cocompact Radical -- 6.1. Hyperplane Subalgebras Of Lie Algebras --^ 6.2. Elementary Facts About Maximal Semigroups -- 6.3. Abelian And Almost Abelian Groups -- 6.4. Nilpotent Groups -- 6.5. Reduction Lemmas -- 6.6. Characterization Of Maximal Subsemigroups -- 6.7. Applications To Reachability Questions -- 7. Invariant Cones And Ol'shanskii Semigroups -- 7.1. Compactly Embedded Cartan Algebras -- 7.2. Invariant Cones In Lie Algebras -- 7.3. Lawson's Theorem On Olshanskii Semigroups -- 8. Compression Semigroups -- 8.1. Invariant Control Sets -- 8.2. Moment Maps And Projective Spaces -- 8.3. Pseudo-unitary Representations And Orbits On Flag Manifolds -- 8.4. Compression Semigroups Of Open G-orbits -- 8.5. Contraction Semigroups For Indefinite Forms -- 8.6. Maximality Of Complex Ol'shanskii Semigroups -- 9. Representation Theory -- 9.1. Involutive Semigroups -- 9.2. Holomorphic Representations Of Half Planes -- 9.3. Invariant Cones And Unitary Representations -- 9.4. Holomorphic Discrete Series Representations -- 9.5. Hardy Spaces --^ 9.6. Howe's Oscillator Semigroup -- 9.7. The Luscher-mack Theorem -- 10. The Theory For Sl(2). Joachim Hilgert, Karl-hermann Neeb. Includes Bibliographical References And Index. Lie semigroups and their tangent wedges....Pages 1-46 Examples....Pages 47-79 Geometry and topology of Lie semigroups....Pages 80-112 Ordered homogeneous spaces....Pages 113-147 Applications of ordered spaces to Lie semigroups....Pages 148-161 Maximal semigroups in groups with cocompact radical....Pages 162-176 Invariant Cones and Ol'shanskii semigroups....Pages 177-201 Compression semigroups....Pages 202-253 Representation theory....Pages 254-296 The theory for Sl(2)....Pages 297-302
