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Holomorphy and convexity in Lie theory

Author: Karl-Hermann Neeb
Publisher: New York : Walter de Gruyter, 2000.
Series: De Gruyter expositions in mathematics, 28.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Neeb, Karl-Hermann.
Holomorphy and convexity in Lie theory
(DLC) 99047514
(OCoLC)42392206
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Karl-Hermann Neeb
ISBN: 9783110808148 3110808145
OCLC Number: 857492977
Description: 1 online resource (xxi, 778 pages).
Contents: A. Abstract Representation Theory --
Chapter I. Reproducing Kernel Spaces 3 --
I.1. Operator-Valued Positive Definite Kernels 3 --
I.2. The Cone of Positive Definite Kernels 14 --
Chapter II. Representations of Involutive Semigroups 20 --
II. 1. Involutive Semigroups 21 --
II. 2. Bounded Representations 24 --
II. 3. Hermitian Representations 29 --
II. 4. Representations on Reproducing Kernel Spaces 34 --
Chapter III. Positive Definite Functions on Involutive Semigroups 52 --
III. 1. Positive Definite Functions --
the Discrete Case 53 --
III. 2. Enveloping C*-algebras 68 --
III. 3. Multiplicity Free Representations 80 --
Chapter IV. Continuous and Holomorphic Representations 99 --
IV. 1. Continuous Representations and Positive Definite Functions 99 --
IV. 2. Holomorphic Representations of Involutive Semigroups 119 --
B. Convex Geometry and Representations of Vector Spaces --
Chapter V. Convex Sets and Convex Functions 125 --
V.1. Convex Sets and Cones 126 --
V.2. Finite Reflection Groups and Convex Sets 138 --
V.3. Convex Functions and Fenchel Duality 147 --
V.4. Laplace Transforms 163 --
V.5. The Characteristic Function of a Convex Set 174 --
Chapter VI. Representations of Cones and Tubes 184 --
VI. 1. Commutative Representation Theory 185 --
VI. 2. Representations of Cones 195 --
VI. 3. Holomorphic Representations of Tubes 205 --
VI. 4. Realization of Cyclic Representations by Holomorphic Functions 209 --
VI. 5. Holomorphic Extensions of Unitary Representations 214 --
C. Convex Geometry of Lie Algebras --
Chapter VII. Convexity in Lie Algebras 221 --
VII. 1. Compactly Embedded Cartan Subalgebras 222 --
VII. 2. Root Decompositions 231 --
VII. 3. Lie Algebras With Many Invariant Convex Sets 251 --
Chapter VIII. Convexity Theorems and Their Applications 265 --
VIII. 1. Admissible Coadjoint Orbits and Convexity Theorems 266 --
VIII. 2. The Structure of Admissible Lie Algebras 292 --
VIII. 3. Invariant Elliptic Cones in Lie Algebras 306 --
D. Highest Weight Representations of Lie Algebras, Lie Groups, and Semigroups --
Chapter IX. Unitary Highest Weight Representations: Algebraic Theory 327 --
IX. 1. Generalized Highest Weight Representations 328 --
IX. 2. Positive Complex Polarizations 344 --
IX. 3. Highest Weight Modules of Finite-Dimensional Lie Algebras 356 --
IX. 4. The Metaplectic Factorization 361 --
IX. 5. Unitary Highest Weight Representations of Hermitian Lie Algebras 374 --
Chapter X. Unitary Highest Weight Representations: Analytic Theory 387 --
X.1. The Convex Moment Set of a Unitary Representation 388 --
X.2. Irreducible Unitary Representations 394 --
X.3. The Metaplectic Representation and Its Applications 400 --
X.4. Special Properties of Unitary Highest Weight Representations 411 --
X.5. Moment Sets for C*-algebras 419 --
X.6. Moment Sets for Group Representations 428 --
Chapter XI. Complex Ol'shanskii Semigroups and Their Representations 442 --
XI. 1. Lawson's Theorem on Ol'shanskii Semigroups 443 --
XI. 2. Holomorphic Extension of Unitary Representations 457 --
XI. 3. Holomorphic Representations of Ol'shanskii Semigroups 464 --
XI. 4. Irreducible Holomorphic Representations 470 --
XI. 5. Gelfand-Raikov Theorems for Ol'shanskii Semigroups 476 --
XI. 6. Decomposition and Characters of Holomorphic Representations 477 --
Chapter XII. Realization of Highest Weight Representations on Complex Domains 493 --
XII. 1. The Structure of Groups of Harish-Chandra Type 494 --
XII. 2. Representations of Groups of Harish-Chandra Type 514 --
XII. 3. The Compression Semigroup and Its Representations 524 --
XII. 5. Hilbert Spaces of Square Integrable Holomorphic Functions 538 --
E. Complex Geometry and Representation Theory --
Chapter XIII. Complex and Convex Geometry of Complex Semigroups 557 --
XIII. 1. Locally Convex Functions and Local Recession Cones 559 --
XIII. 2. Invariant Convex Sets and Functions in Lie Algebras 563 --
XIII. 3. Calculations in Low-Dimensional Cases 571 --
XIII. 4. Biinvariant Plurisubharmonic Functions 576 --
XIII. 5. Complex Semigroups and Stein Manifolds 586 --
XIII. 6. Biinvariant Domains of Holomorphy 595 --
Chapter XIV. Biinvariant Hilbert Spaces and Hardy Spaces on Complex Semigroups 600 --
XIV. 1. Biinvariant Hilbert Spaces 601 --
XIV. 2. Hardy Spaces Defined by Sup-Norms 608 --
XIV. 3. Hardy Spaces Defined by Square Integrability 616 --
XIV. 4. The Fine Structure of Hardy Spaces 623 --
Chapter XV. Coherent State Representations 645 --
XV. 1. Complex Structures on Homogeneous Spaces 646 --
XV. 2. Coherent State Representations 650 --
XV. 3. Heisenberg's Uncertainty Principle and Coherent States 656 --
Appendix I. Bounded Operators on Hilbert Spaces 665 --
Appendix II. Spectral Measures and Unbounded Operators 677 --
Appendix III. Holomorphic Functions on Infinite-Dimensional Spaces 686 --
Appendix IV. Symplectic Geometry 694 --
Appendix V. Simple Modules of p-Length 2 705 --
Appendix VI. Symplectic Modules of Convex Type 715 --
Appendix VII. Square Integrable Representations of Locally Compact Groups 727 --
Appendix VIII. The Stone-von Neumann-Mackey Theorem 742.
Series Title: De Gruyter expositions in mathematics, 28.
Responsibility: by Karl-Hermann Neeb.

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