Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Corrected Second Edition

Contents
Preface
Part I General Theory
1 Matrix Lie Groups
1.1 Definitions
1.2 Examples
1.2.1 General and Special Linear Groups
1.2.2 Unitary and Orthogonal Groups
1.2.3 Generalized Orthogonal and Lorentz Groups
1.2.4 Symplectic Groups
1.2.5 The Euclidean and Poincaré Groups
1.2.6 The Heisenberg Group
1.2.7 The Groups R, C, S1, R, and Rn
1.2.8 The Compact Symplectic Group
1.3 Topological Properties
1.3.1 Compactness
1.3.2 Connectedness
1.3.3 Simple Connectedness
1.3.4 The Topology of SO(3)
1.4 Homomorphisms
1.5 Lie Groups
1.6 Exercises
2 The Matrix Exponential
2.1 The Exponential of a Matrix
2.2 Computing the Exponential
2.3 The Matrix Logarithm
2.4 Further Properties of the Exponential
2.5 The Polar Decomposition
2.6 Exercises
3 Lie Algebras
3.1 Definitions and First Examples
3.2 Simple, Solvable, and Nilpotent Lie Algebras
3.3 The Lie Algebra of a Matrix Lie Group
3.4 Examples
3.5 Lie Group and Lie Algebra Homomorphisms
3.6 The Complexification of a Real Lie Algebra
3.7 The Exponential Map
3.8 Consequences of Theorem 3.42
3.9 Exercises
4 Basic Representation Theory
4.1 Representations
4.2 Examples of Representations
4.3 New Representations from Old
4.3.1 Direct Sums
4.3.2 Tensor Products
4.3.3 Dual Representations
4.4 Complete Reducibility
4.5 Schur's Lemma
4.6 Representations of sl(2;C)
4.7 Group Versus Lie Algebra Representations
4.8 A Nonmatrix Lie Group
4.9 Exercises
5 The Baker–Campbell–Hausdorff Formula and Its Consequences
5.1 The ``Hard'' Questions
5.2 An Illustrative Example
5.3 The Baker–Campbell–Hausdorff Formula
5.4 The Derivative of the Exponential Map
5.5 Proof of the BCH Formula
5.6 The Series Form of the BCH Formula
5.7 Group Versus Lie Algebra Homomorphisms
5.8 Universal Covers
5.9 Subgroups and Subalgebras
5.10 Lie's Third Theorem
5.11 Exercises
Part II Semisimple Lie Algebras
6 The Representations of sl(3;C)
6.1 Preliminaries
6.2 Weights and Roots
6.3 The Theorem of the Highest Weight
6.4 Proof of the Theorem
6.5 An Example: Highest Weight (1,1)
6.6 The Weyl Group
6.7 Weight Diagrams
6.8 Further Properties of the Representations
6.9 Exercises
7 Semisimple Lie Algebras
7.1 Semisimple and Reductive Lie Algebras
7.2 Cartan Subalgebras
7.3 Roots and Root Spaces
7.4 The Weyl Group
7.5 Root Systems
7.6 Simple Lie Algebras
7.7 The Root Systems of the Classical Lie Algebras
7.7.1 The Special Linear Algebras sl(n+1;C)
7.7.2 The Orthogonal Algebras so(2n;C)
7.7.3 The Orthogonal Algebras so(2n+1;C)
7.7.4 The Symplectic Algebras sp(n;C)
7.8 Exercises
8 Root Systems
8.1 Abstract Root Systems
8.2 Examples in Rank Two
8.3 Duality
8.4 Bases and Weyl Chambers
8.5 Weyl Chambers and the Weyl Group
8.6 Dynkin Diagrams
8.7 Integral and Dominant Integral Elements
8.8 The Partial Ordering
8.9 Examples in Rank Three
8.10 The Classical Root Systems
8.10.1 The An Root System
8.10.2 The Dn Root System
8.10.3 The Bn Root System
8.10.4 The Cn Root System
8.10.5 The Classical Dynkin Diagrams
8.11 The Classification
8.12 Exercises
9 Representations of Semisimple Lie Algebras
9.1 Weights of Representations
9.2 Introduction to Verma Modules
9.3 Universal Enveloping Algebras
9.4 Proof of the PBW Theorem
9.5 Construction of Verma Modules
9.6 Irreducible Quotient Modules
9.7 Finite-Dimensional Quotient Modules
9.8 Exercises
10 Further Properties of the Representations
10.1 The Structure of the Weights
10.2 The Casimir Element
10.3 Complete Reducibility
10.4 The Weyl Character Formula
10.5 The Weyl Dimension Formula
10.6 The Kostant Multiplicity Formula
10.7 The Character Formula for Verma Modules
10.8 Proof of the Character Formula
10.9 Exercises
Part III Compact Lie Groups
11 Compact Lie Groups and Maximal Tori
11.1 Tori
11.2 Maximal Tori and the Weyl Group
11.3 Mapping Degrees
11.4 Quotient Manifolds
11.5 Proof of the Torus Theorem
11.6 The Weyl Integral Formula
11.7 Roots and the Structure of the Weyl Group
11.8 Exercises
12 The Compact Group Approach to Representation Theory
12.1 Representations
12.2 Analytically Integral Elements
12.3 Orthonormality and Completeness for Characters
12.4 The Analytic Proof of the Weyl Character Formula
12.5 Constructing the Representations
12.6 The Case in Which δ is Not Analytically Integral
12.7 Exercises
13 Fundamental Groups of Compact Lie Groups
13.1 The Fundamental Group
13.2 Fundamental Groups of Compact Classical Groups
13.3 Fundamental Groups of Noncompact Classical Groups
13.4 The Fundamental Groups of K and T
13.5 Regular Elements
13.6 The Stiefel Diagram
13.7 Proofs of the Main Theorems
13.8 The Center of K
13.9 Exercises
Erratum
A Linear Algebra Review
A.1 Eigenvectors and Eigenvalues
A.2 Diagonalization
A.3 Generalized Eigenvectors and the SN Decomposition
A.4 The Jordan Canonical Form
A.5 The Trace
A.6 Inner Products
A.7 Dual Spaces
A.8 Simultaneous Diagonalization
B Differential Forms
C Clebsch–Gordan Theory and the Wigner–Eckart Theorem
C.1 Tensor Products of sl(2;C) Representations
C.2 The Wigner–Eckart Theorem
C.3 More on Vector Operators
D Peter–Weyl Theorem and Completeness of Characters
References
Index