緊李羣

Preface

1 Compact Lie Groups

1.1 Basic Notions

1.1.1 Manifolds

1.1.2 Lie Groups

1.1.3 Lie Subgroups and Homomorphisms

1.1.4 Compact Classical Lie Groups

1.1.5 Exercises

1.2 Basic Topology

1.2.1 Connectedness

1.2.2 Simply Connected Cover

1.2.3 Exercises

1.3 The Double Cover of SO(n)

1.3.1 Clifford Algebras

1.3.2 Spinn(IR) and Pin

1.3.3 Exercises

1.4 Integration

1.4.1 Volume Forms

1.4.2 Invafiant Integration

1.4.3 Fubini's Theorem

1.4.4 Exercises

2 Representations

2.1 Basic Notions

2.1.1 Definitions

2.1.2 Examples

2.1.3 Exercises

2.2 Operations on Representations

2.2.1 Constructing New Representations

2.2.2 Irreducibility and Schur's Lemma

2.2.3 Unitarity

2.2.4 Canonical Decomposition

2.2.5 Exercises

2.3 Examples of Irreducibility

2.3.1 SU(2) and Vn(C2)

2.3.2 SO(n) and Harmonic Polynomials

2.3.3 Spin and Half-Spin Representations

2.3.4 Exercises

3 Harmonic Analysis

3.1 Matrix Coefficients

3.1.1 Schur Orthogonality

3.1.2 Characters

3.1.3 Exercises

3.2 Infinite-Dimensional Representations

3.2.1 Basic Definitions and Schur's Lemma

3.2.2 G-Finite Vectors

3.2.3 Canonical Decomposition

3.2.4 Exercises

3.3 The Peter-Weyl Theorem

3.3.1 The Left and Right Regular Representation

3.3.2 Main Result

3.3.3 Applications

3.3.4 Exercises

3.4 Fourier Theory

3.4.1 Convolution

3.4.2 Plancherel Theorem

3.4.3 Projection Operators and More General Spaces

3.4.4 Exercises

4 Lie Algebras

4.1 Basic Definitions

4.1.1 Lie Algebras of Linear Lie Groups

4.1.2 Exponential Map

4.1.3 Lie Algebras for the Compact Classical Lie Groups

4.1.4 Exercises

4.2 Further Constructions

4.2.1 Lie Algebra Homomorphisms

4.2.2 Lie Subgroups and Subalgebras

4.2.3 Covering Homomorphisms

4.2.4 Exercises

5 Abelian Lie Subgroups and Structure

5.1 Abelian Subgroups and Subalgebras

5.1.1 Maximal Tori and Caftan Subalgebras

5.1.2 Examples

5.1.3 Conjugacy of Cartan Subalgehras

5.1.4 Maximal Torus Theorem

5.1.5 Exercises

5.2 Structure

5.2.1 Exponential Map Revisited

5.2.2 Lie Algebra Structure

5.2.3 Commutator Theorem

5.2.4 Compact Lie Group Structure

5.2.5 Exercises

6 Roots and Associated Structures

6.1 Root Theory

6.1.1 Representations of Lie Algebras

6.1.2 Complexification of Lie Algebras

6.1.3 Weights

6.1.4 Roots

6.1.5 Compact Classical Lie Group Examples

6.1.6 Exercises

6.2 The Standard sl(2, C) Triple

6.2.1 Cartan Involution

6.2.2 Killing Form

6.2.3 The Standard sl(2, C) and su(2) Triples

6.2.4 Exercises

6.3 Lattices

6.3.1 Definitions

6.3.2 Relations

6.3.3 Center and Fundamental Group

6.3.4 Exercises

6.4 Weyl Group

6.4.1 Group Picture

6.4.2 Classical Examples

6.4.3 Simple Roots and Weyl Chambers

6.4.4 The Weyl Group as a Reflection Group

6.4.5 Exercises

7 Highest Weight Theory

7.1 Highest Weights

7.1.1 Exercises

7.2 Weyl Integration Formula

7.2.1 Regular Elements

7.2.2 Main Theorem

7.2.3 Exercises

7.3 Weyl Character Formula

7.3.1 Machinery

7.3.2 Main Theorem

7.3.3 Weyl Denominator Formula

7.3.4 Weyl Dimension Formula

7.3.5 Highest Weight Classification

7.3.6 Fundamental Group

7.3.7 Exercises

7.4 Borel-Weil Theorem

7.4.1 Induced Representations

7.4.2 Complex Structure on G/T

7.4.3 Holomorphic Functions

7.4.4 Main Theorem

7.4.5 Exercises

References

Index ^[1]