Quantum Linear Groups

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Quantum linear groups

Copyright (c) 1991, American Mathematical Society
ISBN 0-8218-2501-1
QA171.P372 1991 512'.2-dc20
LCCN 90019310

Contents

Abstract

Introduction

Chapter 1. Quantum Groups
1.1. Quantum affine spaces
1.2. Quantum groups
1.3. Direct products
1.4. Closed subgroups
1.5. Normal closed subgroups
1.6. Kernels and exact sequences
1.7. Cartesian squares
1.8. Coverings

Chapter 2. Representation Theory of Quantum Groups
2.1. Rational representations.
2.2. Functorial description
2.3. Defining matrices
2.4. Contragredient modules and tensor products
2.5. Characters and character groups
2.6. Fixed points
2.7. Induction.
2.8. Injective objects
2.9. Exact subgroups of quantum groups
2.10. A theorem on central faithfully flat morphisms
2.11. The Hochschild-Serre spectral sequence

Chapter 3. Quantum Linear Spaces and Quantum Matrix Spaces
3.1. Quadratic algebras
3.2. Quasi-Yang-Baxter algebras
3.3. Basis theorem for quasi-Yang-Baxter algebras
3.4. Quadratic algebras K[AQ 1°] and K[AQ1n].
3.5. Quantum matrix space M, (n).
3.6. The bialgebra structure on K [Mq (n)].
3.7. Some automorphisms and anti-automorphisms

Chapter 4. Quantum Determinants
4.1. Quantum determinant
4.2. First properties of the determinant.
4.3. Sub determinants
4.4. Laplace expansions
4.5. Some commutators, I.
4.6. The centrality of the determinant

Chapter 5. Antipode and Quantum Linear Groups
5.1. Some commutators, II.
5.2. Some commutators, III.
5.3. Quantum general and special linear groups
5.4. A property of the antipode

Chapter 6. Some Closed Subgroups
6.1. Parabolic and Levi subgroups.
6.2. Some properties of the parabolic and Levi subgroups.
6.3. Some remarks
6.4. Coadjoint action of the maximal torus and the root system.
6.5. Character groups of T. and Bq.

Chapter 7. Frobenius Morphisms and Kernels
7.1. Gaussian polynomials
7.2. Frobenius morphisms
7.3. Infinitesimal subgroups
7.4. Some homological properties of GLq(n).
7.5. Some exact subgroups of GLq (n).

Chapter 8. Global Representation Theory
8.1. Density of the "big cell"
8.2. Highest weight modules
8.3. Some properties of induced Gq-modules
8.4. Induction to parabolic subgroups
8.5. The semisimple rank 1 case, I.
8.6. The semisimple rank 1 case, II.
8.7. The one-to-one correspondence between irreducible modules and dominant weights.
8.8. Formal characters and their invariance under the Weyl group
8.9. Injective modules for Borel subgroups
8.10. A finiteness theorem; Weyl modules

Chapter 9. Infinitesimal Representation Theory
9.1. An infinitesimal version of the "density theorem".
9.2. Highest weight and irreducible representations of (Gq)i T and (G, )l B.
9.3. Irreducible representations of (Gq )1.
9.4. The tensor product theorem
9.5. Induction to "infinitesimal Borel subgroups".
9.6. Induction from "infinitesimal Borel subgroups", I.
9.7. Induction from "infinitesimal Borel subgroups", II.
9.8. Highest weight categories
9.9. Injective modules for (Gq)i.
9.10. The Steinberg module.

Chapter 10. The Generalization of Certain Important Theorems on the Cohomology of Vector Bundles on the Flag Manifold
10.1. An isomorphism theorem and its consequences
10.2. Borel-Weil-l3ott theorem for small dominant weights.
10.3. Serre duality and strong linkage principle
10.4. Kempf vanishing theorem, good filtrations and Weyl character formula.
10.5. A coalgebra isomorphism between K [GLq (n)] and K [GL _ q (n )] .

Chapter 11. q-Schur AlgebrasIn
11.1. Polynomial representations of Gq.
11.2. q-Schur algebras
11.3. Sq(n, r) as an endomorphism algebra
11.4. On the complete reducibility of Gq-modules.
11.5. S. (n, r) as a quasi-hereditary algebra.
11.6. The generalization of a theorem of J. A. Green
11.7. Tensor product theorem for q-Schur algebras.

References

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