This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory.
These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.
Author(s): Anthony W. Knapp
Series: MN-34 Mathematical Notes
Publisher: Princeton University Press
Year: 1988
Language: English
Pages: 518
Contents......Page 5
Preface......Page 7
1. SO(3) and its Lie algebra......Page 12
2. Exponential of a matrix......Page 16
3. Closed linear groups......Page 19
4. Manifolds and Lie groups......Page 24
5. Closed linear groups as Lie groups......Page 27
6. Homomorphisms......Page 34
7. An interesting homomorphism......Page 40
8. Representations......Page 45
1. Abstract Lie algebras......Page 54
2. Tensor product of two representations......Page 57
3. Representations on the tensor algebra......Page 65
4. Representations on exterior and symmetric algebras......Page 73
5. Extension of scalars - complexification......Page 82
6. Universal enveloping algebra......Page 84
7. Symmetrization......Page 95
8. Tensor products over an algebra......Page 101
1. Abstract theory......Page 108
2. Irreducible representations of SU(2)......Page 120
3. Root space decomposition for U(n)......Page 123
4. Roots and weights for U(n)......Page 127
5. Theorem of the Highest Weight for U(n)......Page 133
6. Weyl group for U(n)......Page 138
7. Analytic form of Borel-Weil Theorem for U(n)......Page 147
1. Motivation from differential forms......Page 162
2. Motivation from extensions......Page 170
3. Definition and examples......Page 175
4. Computation from any free resolution......Page 184
5. Lemmas for Koszul resolution......Page 197
6. Exactness of Koszul resolution......Page 199
1. Projectives and injectives......Page 208
2. Functors......Page 219
3. Derived functors......Page 239
4. Connecting homomorphisms and long exact sequences......Page 247
5. Long exact sequence for derived functors......Page 254
6. Naturality of long exact sequence......Page 267
1. Projectives and injectives......Page 275
2. Lie algebra homology and cohomology......Page 291
3. Poincaré duality......Page 297
4. Kostant's Theorem for U(n)......Page 301
5. Harish-Chandra isomorphism for U(n)......Page 310
6. Casselman-Osborne Theorem......Page 326
1. Motivation for how to construct representations......Page 334
2. (g,K) modules......Page 343
3. The algebra R(g,K)......Page 349
4. The category C(g,k)......Page 369
5. The functors P and I......Page 374
6. Projectives and injectives......Page 382
7. Homology, cohomology, and Ext......Page 393
8. Standard resolutions......Page 395
9. Poincaré duality......Page 405
10. Revised setting for Kostant's Theorem......Page 411
11. Borel-Weil-Bott Theorem for U(n)......Page 417
1. Structure theory for U(m,n)......Page 426
2. Cohomological induction......Page 437
3. Vanishing above the middle dimension......Page 446
4. First reduction below the middle dimension......Page 457
5. Second reduction below the middle dimension......Page 469
6. Vanishing below the middle dimension......Page 477
7. Effect on infinitesimal character......Page 482
8. Effect on multiplicities of K types......Page 488
Notes......Page 499
References......Page 509
Index of Notation......Page 512
Index......Page 515
