The Structure of Compact Groups A Primer for the Student A Handbook for the Expert 3rd Edition by Karl H Hofmann, Sidney A Morris ISBN 3110296551 9783110296556

Chapter 1. Basic Topics and Examples

Definitions and Elementary Examples

Actions, Subgroups, Quotient Spaces

Products of Compact Groups

Applications to Abelian Groups

Projective Limits

Totally Disconnected Compact Groups

Some Duality Theory

Postscript

References for this Chapter—Additional Reading

Chapter 2. The Basic Representation Theory of Compact Groups

Some Basic Representation Theory for Compact Groups

The Haar Integral

Consequences of Haar Measure

The Main Theorem on Hilbert Modules for Compact Groups

Postscript

References for this Chapter—Additional Reading

Chapter 3. The Ideas of Peter and Weyl

Part 1: The Classical Theorem of Peter and Weyl

An Excursion into Linear Algebra

The G-modules E’ ⊗ E, Hom(E,E) and Hom(E,E)’

The Fine Structure of R(G,K)

Part 2: The General Theory of G-Modules

Vector Valued Integration

The First Application: The Averaging Operator

Compact Groups Acting on Convex Cones

More Module Actions, Convolutions

Complexification of Real Representations

Postscript

References for this Chapter—Additional Reading

Chapter 4. Characters

Part 1: Characters of Finite Dimensional Representations

Part 2: The Structure Theorem of Efin

Cyclic Modules

Postscript

References for this Chapter—Additional Reading

Chapter 5. Linear Lie Groups

Preliminaries

The Exponential Function and the Logarithm

Differentiating the Exponential Function in a Banach Algebra

Local Groups for the Campbell–Hausdorff Multiplication

Subgroups of A-1 and Linear Lie Groups

Analytic Groups

The Intrinsic Exponential Function of a Linear Lie Group

The Adjoint Representation of a Linear Lie Group

Subalgebras, Ideals, Lie Subgroups, Normal Lie Subgroups

Normalizers, Centralizers, Centers

The Commutator Subgroup

Forced Continuity of Morphisms between Lie Groups

Quotients of Linear Lie Groups

The Topological Splitting Theorem for Normal Vector Subgroups

Postscript

References for this Chapter—Additional Reading

Chapter 6. Compact Lie Groups

Compact Lie algebras

The Commutator Subgroup of a Compact Lie Group

The Structure Theorem for Compact Lie Groups

Maximal Tori

The Second Structure Theorem for Connected Compact Lie Groups

Compact Abelian Lie Groups and their Linear Actions

Action of a Maximal Torus on the Lie Algebra

The Weyl Group Revisited

The Commutator Subgroup of Connected Compact Lie Groups

On the Automorphism Group of a Compact Lie Group

Covering Groups of Disconnected Compact Lie Groups

Auerbach’s Generation Theorem

The Topology of Connected Compact Lie Groups

Postscript

References for this Chapter—Additional Reading

Chapter 7. Duality of Abelian Topological Groups

The Compact Open Topology and Hom-Groups

Local Compactness and Duality of Abelian Topological Groups

Basic Functorial Aspects of Duality

The Annihilator Mechanism

Character Groups of Topological Vector Spaces

The Exponential Function

Weil’s Lemma and Compactly Generated Abelian Groups

Reducing Locally Compact Groups to Compact Abelian Groups

A Major Structure Theorem

The Duality Theorem

The Identity Component

The Weight of Locally Compact Abelian Groups

Postscript

References for this Chapter—Additional Reading

Chapter 8. Compact Abelian Groups

Part 1: Aspects of the Algebraic Structure

Divisibility, Torsion, Connectivity

Compact Abelian Groups as Factor Groups

Part 2: Aspects of the Point Set Topological Structure

Topological Dimension of Compact Abelian Groups

Arc Connectivity

Local Connectivity

Compact Metric Abelian Groups

Part 3: Aspects of Algebraic Topology—Homotopy

Free Compact Abelian Groups

Homotopy of Compact Abelian Groups

Exponential Function and Homotopy

The Fine Structure of Free Compact Abelian Groups

Part 4: Aspects of Homological Algebra

Injective, Projective, and Free Compact Abelian Groups

Part 5: Aspects of Algebraic Topology—Cohomology

Cohomology of Compact Abelian Groups

Part 6: Aspects of Set Theory

Arc Components and Borel Subsets

Postscript

References for this Chapter—Additional Reading

Chapter 9. The Structure of Compact Groups

Part 1: The Fundamental Structure Theorems of Compact Groups

Approximating Compact Groups by Compact Lie Groups

The Closedness of Commutator Subgroups

Semisimple Compact Connected Groups

The Levi–Mal’cev Structure Theorem for Compact Groups

Maximal Connected Abelian Subgroups

The Splitting Structure Theorem

Supplementing the Identity Component

Part 2: The Structure Theorems for the Exponential Function

The Exponential Function of Compact Groups

The Dimension of Compact Groups

Locally Euclidean Compact Groups Are Compact Lie Groups

Part 3: The Connectivity Structure of Compact Groups

Arc Connectivity

Local Connectivity

Compact Groups and Indecomposable Continua

Part 4: Some Homological Algebra for Compact Groups

The Projective Cover of Connected Compact Groups

Part 5: The Automorphism Group of Compact Groups

The Iwasawa Theory of Automorphism Groups

Simple Compact Groups and the Countable Layer Theorem

The Structure of Compact FC-Groups

The Commutativity Degree of a Compact Group

Postscript

References for this Chapter—Additional Reading

Chapter 10. Compact Group Actions

A Preparation Involving Compact Semigroups

Orbits, Orbit Space, and Isotropy

Equivariance and Cross Sections

Triviality of an Action

Quotient Actions, Totally Disconnected G-Spaces

Compact Lie Group Actions on Locally Compact Spaces

Triviality Theorems for Compact Group Actions

Split Morphisms

Actions of Compact Groups and Acyclicity

Fixed Points of Compact Abelian Group Actions

Transitive Actions of Compact Groups

Szenthe’s Theory of Transitive Actions of Compact Groups

Postscript

References for this Chapter—Additional Reading

Chapter 11. The Structure of Free Compact Groups

The Category Theoretical Background

Splitting the Identity Component

The Center of a Free Compact Group

The Commutator Subgroup of a Free Compact Group

Freeness Versus Projectivity

Postscript

References for this Chapter—Additional Reading

Chapter 12. Cardinal Invariants of Compact Groups

Suitable Sets

Generating Degree and Density

The Cardinal Invariants of Connected Compact Groups

Cardinal Invariants in the Absence of Connectivity

On the Location of Special Generating Sets

Postscript

References for this Chapter—Additional Reading

Appendix 1. Abelian Groups

Examples

Free Abelian Groups

Projective Groups

Torsion Subgroups

Pure Subgroups

Free Quotients

Divisibility

Some Homological Algebra

Exact Sequences

Whitehead’s Problem

Postscript

References for this Appendix—Additional Reading

Appendix 2. Covering Spaces and Groups

Covering Spaces and Simple Connectivity

The Group of Covering Transformations

Universal Covering Groups

Groups Generated by Local Groups

Postscript

References for this Appendix—Additional Reading

Appendix 3. A Primer of Category Theory

Categories, Morphisms

Pointed Categories

Types of Morphisms

Functors

Natural Transformations

Equivalence of Categories

Limits

The Continuity of Adjoints

The Left Adjoint Existence Theorem

Commutative Monoidal Categories and its Monoid Objects

Part 1: The Quintessential Diagram Chase

Part 2: Connected Graded Commutative Hopf Algebras

Part 3: Duality of Graded Hopf Algebras

Part 4: An Application to Compact Monoids

Postscript

References for this Appendix—Additional Reading

Appendix 4. Selected Results on Topology and Topological Groups

The Arc Component Topology

The Weight of a Topological Space

Metrizability of Topological Groups

Duality of Vector Spaces

Subgroups of Topological Groups

Wallace’s Lemma

Cantor Cubes and Dyadic Spaces

Some Basic Facts on Compact Monoids

Postscript

References for this Appendix—Additional Reading

Appendix 5. Measures on Compact Groups

The Definition of Haar Measure

The Required Background of Radon Measure Theory

Product Measures

The Support of a Measure

Measures on Compact Groups: Convolution

Semigroup Theoretical Characterization of Haar Measure

Idempotent Probability Measures on a Compact Group

Actions and Product Measures

Postscript

References for this Appendix—Additional Reading

Appendix 6. Projective Limits of Well-Ordered Inverse Systems

Well-ordered Lie chains

Supercompactness

Compact Homeomorphism Groups

Postscript

References for this Appendix—Additional Reading

References

Index of Symbols

Index