Elements of the Theory of Representations (Paperback, Softcover Repri)

'First Part. Preliminary Facts.- 1. Sets, Categories, Topology.- 1.1. Sets.- 1.2. Categories and Functors.- 1.3. The Elements of Topology.- 2. Groups and Homogeneous Spaces.- 2.1. Transformation Groups and Abstract Groups.- 2.2. Homogeneous Spaces.- 2.3. Principal Types of Groups.- 2.4. Extensions of Groups.- 2.5. Cohomology of Groups.- 2.6. Topological Groups and Homogeneous Spaces.- 3. Rings and Module.- 3.1. Rings.- 3.2. Skew Fields.- 3.3. Modules over Rings.- 3.4. Linear Spaces.- 3.5. Algebra.- 4. Elements of Functional Analysis.- 4.1. Linear Topological Spaces.- 4.2. Banach Algebras.- 4.3. C -Algebras.- 4.4. Commutative Operator Algebras.- 4.5. Continuous Sums of Hilbert Spaces and von Neumann Algebras.- 5. Analysis on Manifolds.- 5.1. Manifolds.- 5.2. Vector Fields.- 5.3. Differential Forms.- 5.4. Bundles.- 6. Lie Groups and Lie Algebras..- 6.1. Lie Group.- 6.2. Lie Algebras.- 6.3. The Connection between Lie Groups and Lie Algebras.- 6.4. The Exponential Mapping.- Second Part. Basic Concepts and Methods of the Theory of Representations.- 7. Representations of Groups.- 7.1. Linear Representations.- 7.2. Representations of Topological Groups in Linear Topological Space.- 7.3. Unitary Representations.- 8. Decomposition of Representations.- 8.1. Decomposition of Finite Representation.- 8.2. Irreducible Representation.- 8.3. Completely Reducible Representations.- 8.4. Decomposition of Unitary Representations.- 9. Invariant Integration.- 9.1. Means and Invariant Measures.- 9.2. Applications to Compact Groups.- 9.3. Applications to Noncompact Groups.- 10. Group Algebras.- 10.1. The Group Ring ofa Finite Group.- 10.2. Group Algebras of Topological Groups.- 10.3. Application of Group C -Algebras.- 10.4. Group Algebras of Lie Groups.- 10.5. Representations of Lie Groups and their Group Algebras.- 11. Characters..- 11.1. Characters of Finite-Dimensional Representations.- 11.2. Characters of Infinite-Dimensional Representations.- 11.3. Infinitesimal Characters.- 12. Fourier Transforms and Duality.- 12.1. Commutative Groups.- 12.2. Compact Groups.- 12.3. Ring Groups and Duality for Finite Groups.- 12.4. Other Result.- 13. Induced Representations.- 13.1. Induced Representations of Finite Groups.- 13.2. Unitary Induced Representations of Locally Compact Groups.- 13.3. Representations of Group Extensions.- 13.4. Induced Representations of Lie Groups and their Generalizations.- 13.5. Intertwining Operators and Duality.- 13.6. Characters of Induced Representations.- 14. Projective Representation.- 14.1. Projective Groups and Projective Representations.- 14.2. Schur's Theory.- 14.3. Projective Representations of Lie Group.- 15. The Method of Orbits.- 15.1. The Co-Adjoint Representation of a Lie Group.- 15.2. Homogeneous Symplectic Manifold.- 15.3. Construction of an Irreducible Unitary Representation by an Orbit.- 15.4. The Method of Orbits and Quantization of Hamiltonian Mechanical System.- 15.5. Functorial Properties of the Correspondence between Orbits and Representation.- 15.6. The Universal Formula for Characters and Plancherel Measures.- 15.7. Infinitesimal Characters and Orbits.- Third Part. Various Examples.- 16. Finite Groups.- 16.1. Harmonic Analysis on the Three-Dimensional Cube.- 16.1. Harmonic Analysis on the Three-Dimensional Cube.- 16.3. Representations of the Group SL(2,Fq).- 16.4. Vector Fields on Spheres.- 17. Compact Groups.- 17.1. Harmonic Analysis on the Sphere.- 17.2. Representations of the Classical Compact Lie Groups.- 17.3. Spinor Representations of the Orthogonal Group.- 18. Lie Groups and Lie Algebras.- 18.1. Representations of a Simple Three-Dimensional Lie Algebra.- 18.2. The Weyl Algebra and Decomposition of Tensor Products.- 18.3. The Structure of the Enveloping Algebra $$ U\left( \mathfrak{g} ight) $$ for $$ \mathfrak{g} = \mathfrak{s}\mathfrak{l}\left( {2,C} ight) $$.- 18.4. Spinor Representations of the S