Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. the book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.
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目录
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(R)and Pinn(R)
1.3.3 Exercises
1.4 Integration
1.4.1 Volume Forms
1.4.2 Invariant 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