Lie groups, Lie algebras, and cohomology

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Title page Preface Chapter I. Lie Groups and Lie Algebras 1. SO(3) and its Lie algebra 2. Exponential of a matrix 3. Closed linear groups 4. Manifolds and Lie groups 5. Closed linear groups as Lie groups 6. Homomorphisms 7. An interesting homomorphism 8. Representations Chapter II. Representations and Tensors 1. Abstract Lie algebras 2. Tensor product of two representations 3. Representations on the tensor algebra 4. Representations on exterior and symmetric algebras 5. Extension of scalars - complexification 6. Universal enveloping algebra 7. Symmetrization 8. Tensor products over an algebra Chapter III. Representations of Compact Groups 1. Abstract theory 2. Irreducible representations of SU(2) 3. Root space decomposition for U(n) 4. Roots and weights for U(n) 5. Theorem of the Highest Weight for U(n) 6. Weyl group for U(n) 7. Analytic form of Borel-Weil Theorem for U(n) Chapter IV. Cohomology of Lie Algebras 1. Motivation from differential forms 2. Motivation from extensions 3. Definition and examples 4. Computation from any free resolution 5. Lemmas for Koszul resolution 6. Exactness of Koszul resolution Chapter V. Homological Algebra 1. Projectives and injectives 2. Functors 3. Derived functors 4. Connecting homomorphisms and long exact sequences 5. Long exact sequence for derived functors 6. Naturality of long exact sequence Chapter VI. Application to Lie Algebras 1. Projectives and injectives 2. Lie algebra homology and cohomology 3. Poincaré duality 4. Kostant's Theorem for U(n) 5. Harish-Chandra isomorphism for U(n) 6. Casselman-Osborne Theorem Chapter VII. Relative Lie Algebra Cohomology 1. Motivation for how to construct representations 2. (g,K) modules 3. The algebra R(g,K) 4. The category C(g,k) 5. The functors P and I 6. Projectives and injectives 7. Homology, cohomology, and Ext 8. Standard resolutions 9. Poincaré duality 10. Revised setting for Kostant's Theorem 11. Borel-Weil-Bott Theorem for U(n) Chapter VIII. Representations of Noncompact Groups 1. Structure theory for U(m,n) 2. Cohomological induction 3. Vanishing above the middle dimension 4. First reduction below the middle dimension 5. Second reduction below the middle dimension 6. Vanishing below the middle dimension 7. Effect on infinitesimal character 8. Effect on multiplicities of K types Notes References Index of Notation Index