viii CONTENT S
3.6. The Group Algebra 48
3.7. The Measure Algebra 50
3.8. Adele Groups 51
Part 2.
REPRESENTATION THEORY AND COMPACT GROUPS
Chapter 4. Basic Representation Theory 55
4.1. Definitions and Examples 56
4.2. Subrepresentations and Quotient Representations 59
4.3. Operations on Representations 64
4.3A. Dual Space 64
4.3B. Direct Sum 64
4.3C. Tensor Product of Spaces 65
4.3D. Horn 67
4.3E. Bilinear Forms 67
4.3F. Tensor Products of Algebras 68
4.3G. Relation with the Commuting Algebra 69
4.4. Multiplicities and the Commuting Algebra 70
4.5. Completely Continuous Representations 72
4.6. Continuous Direct Sums of Representations 75
4.7. Induced Representations 77
4.8. Vector Bundle Interpretation 81
4.9. Mackey's Little-Group Theorem 82
4.9A. The Normal Subgroup Case 82
4.9B. Cohomology and Projective Representations 84
4.9C. Cocycle Representations and Extensions 85
4.10. Mackey Theory and the Heisenberg Group 87
93
93
96
97
99
101
104
107
107
107
110
111
111
112
113
115
Chapter
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.9.
5. Representations of Compact Groups
Finite Dimensionality
Orthogonality Relations
Characters and Projections
The Peter-Weyl Theorem
The Plancherel Formula
Decomposition into Irreducibles
Some Basic Examples
5.7A. The Group 17(1)
5.7B. The Group SU(2)
5.7C. The Group SO(3)
5.7D. The Group 50(4)
5.7E. The Sphere
S2
5.7F. The Sphere S3
Real, Complex and Quaternion Representations
The Frobenius Reciprocity Theorem
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