Holomorphy and Convexity in Lie Theory

Karl-Hermann Neeb

• Publisher: De Gruyter
• Year: 2011
• Print ISBN:
  9783110156690
• eBook ISBN:
  9783110808148
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

No detailed description available for "Holomorphy and Convexity in Lie Theory".

Table of contents

• Preface
• Introduction
• A. Abstract Representation Theory
• Chapter I. Reproducing Kernel Spaces
• I.1. Operator-Valued Positive Definite Kernels
• I.2. The Cone of Positive Definite Kernels
• Chapter II. Representations of Involutive Semigroups
• II.1. Involutive Semigroups
• II.2. Bounded Representations
• II.3. Hermitian Representations
• II.4. Representations on Reproducing Kernel Spaces
• Chapter III. Positive Definite Functions on Involutive Semigroups
• III.1. Positive Definite Functions — the Discrete Case
• III.2. Enveloping C*-algebras
• III.3. Multiplicity Free Representations
• Chapter IV. Continuous and Holomorphic Representations
• IV.1. Continuous Representations and Positive Definite Functions
• IV.2. Holomorphic Representations of Involutive Semigroups
• B. Convex Geometry and Representations of Vector Spaces
• Chapter V. Convex Sets and Convex Functions
• V.1. Convex Sets and Cones
• V.2. Finite Reflection Groups and Convex Sets
• V.3. Convex Functions and Fenchel Duality
• V.4. Laplace Transforms
• V.5. The Characteristic Function of a Convex Set
• Chapter VI. Representations of Cones and Tubes
• VI.1. Commutative Representation Theory
• VI.2. Representations of Cones
• VI.3. Holomorphic Representations of Tubes
• VI.4. Realization of Cyclic Representations by Holomorphic Functions
• VI.5. Holomorphic Extensions of Unitary Representations
• C. Convex Geometry of Lie Algebras
• Chapter VII. Convexity in Lie Algebras
• VII.1. Compactly Embedded Cartan Subalgebras
• VII.2. Root Decompositions
• VII.3. Lie Algebras With Many Invariant Convex Sets
• Chapter VIII. Convexity Theorems and Their Applications
• VIII.1. Admissible Coadjoint Orbits and Convexity Theorems
• VIII.2. The Structure of Admissible Lie Algebras
• VIII.3. Invariant Elliptic Cones in Lie Algebras
• D. Highest Weight Representations of Lie Algebras, Lie Groups, and Semigroups
• Chapter IX. Unitary Highest Weight Representations: Algebraic Theory
• IX.1. Generalized Highest Weight Representations
• IX.2. Positive Complex Polarizations
• IX.3. Highest Weight Modules of Finite-Dimensional Lie Algebras
• IX.4. The Metaplectic Factorization
• IX.5. Unitary Highest Weight Representations of Hermitian Lie Algebras
• Chapter X. Unitary Highest Weight Representations: Analytic Theory
• X.1. The Convex Moment Set of a Unitary Representation
• X.2. Irreducible Unitary Representations
• X.3. The Metaplectic Representation and Its Applications
• X.4. Special Properties of Unitary Highest Weight Representations
• X.5. Moment Sets for C*-algebras
• X.6. Moment Sets for Group Representations
• Chapter XI. Complex Ol’shanskiĭ Semigroups and Their Representations
• XI.1. Lawson’s Theorem on Ol’shanskiĭ Semigroups
• XI.2. Holomorphic Extension of Unitary Representations
• XI.3. Holomorphic Representations of Ol’shanskiĭ Semigroups
• XI.4. Irreducible Holomorphic Representations
• XI.5. Gelfand-Raïkov Theorems for Ol’shanskiĭ Semigroups
• XI.6. Decomposition and Characters of Holomorphic Representations
• Chapter XII. Realization of Highest Weight Representations on Complex Domains
• XII.1. The Structure of Groups of Harish-Chandra Type
• XII.2. Representations of Groups of Harish-Chandra Type
• XII.3. The Compression Semigroup and Its Representations
• XII.4. Examples
• XII.5. Hilbert Spaces of Square Integrable Holomorphic Functions
• E. Complex Geometry and Representation Theory
• Chapter XIII. Complex and Convex Geometry of Complex Semigroups
• XIII.1. Locally Convex Functions and Local Recession Cones
• XIII.2. Invariant Convex Sets and Functions in Lie Algebras
• XIII.3. Calculations in Low-Dimensional Cases
• XIII.4. Biinvariant Plurisubharmonic Functions
• XIII.5. Complex Semigroups and Stein Manifolds
• XIII.6. Biinvariant Domains of Holomorphy
• Chapter XIV. Biinvariant Hilbert Spaces and Hardy Spaces on Complex Semigroups
• XIV.1. Biinvariant Hilbert Spaces
• XIV.2. Hardy Spaces Defined by Sup-Norms
• XIV.3. Hardy Spaces Defined by Square Integrability
• XIV.4. The Fine Structure of Hardy Spaces
• Chapter XV. Coherent State Representations
• XV.1. Complex Structures on Homogeneous Spaces
• XV.2. Coherent State Representations
• XV.3. Heisenberg’s Uncertainty Principle and Coherent States
• Appendices
• Appendix I. Bounded Operators on Hilbert Spaces
• Appendix II. Spectral Measures and Unbounded Operators
• Appendix III. Holomorphic Functions on Infinite-Dimensional Spaces
• Appendix IV. Symplectic Geometry
• Appendix V. Simple Modules of p-Length 2
• Appendix VI. Symplectic Modules of Convex Type
• Appendix VII. Square Integrable Representations of Locally Compact Groups
• Appendix VIII. The Stone – von Neumann-Mackey Theorem
• Bibliography
• List of Symbols
• Index