Quantum Linear Groups.

Intro -- Contents -- Introduction -- 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 -- 2. Representation Theory of Quantum Groups -- 2.1. Rational representations -- 2.2. Functorial description -- 2.3. Defining matrices -- 2.4. Contragradient 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 -- 3. Quantum Matrix Spaces -- 3.1. Quadratic algebras -- 3.2. Quasi-Yang-Baxter algebras -- 3.3. Basis theorem for quasi-Yang-Baxter algebras -- 3.4. The quadratic algebras K[A[sup(n‌0)][sub(q)]] and K[A[sup(n‌0)][sub(q)]] -- 3.5. The quantum matrix space M[sub(q)](n) -- 3.6. The bialgebra structure on K[M[sub(q)](n)] -- 3.7. Some automorphisms and anti-automorphisms -- 3.8. K[A[sup(n‌0)][sub(q)]] and K[A[sup(n‌0)][sub(q)] as K[M[sub(q)](n)]-comodules -- 4. Quantum Determinants -- 4.1. Quantum determinant -- 4.2. First properties of the determinant -- 4.3. Subdeterminants -- 4.4. Laplace expansions -- 4.5. Some commutators, I -- 4.6. The centrality of the determinant -- 5. The 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 -- 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[sub(q)] and B[sub(q)] -- 7. Frobenius Morphisms and Kernels.

7.1. Gaussian polynomials -- 7.2. Frobenius morphisms -- 7.3. Infinitesimal subgroups -- 7.4. Some homological properties of GL[sub(q)](n) -- 7.5. Some exact subgroups of GL[sub(q)](n) -- 8. Global Representation Theory -- 8.1. Density of the "big cell -- 8.2. Highest weight modules -- 8.3. Some properties of induced G[sub(q)]-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 -- 9. Infinitesimal Representation Theory -- 9.1. An infinitesimal version of the "density theorem -- 9.2. Highest weight and irreducible representations for (G[sub(q)])[sub(1)]-T and (G[sub(q)])[sub(1)]-B -- 9.3. Irreducible representations of (G[sub(q)])[sub(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 (G[sub(q)])[sub(1)] -- 9.10. The Steinberg module -- 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-Bott 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[GL[sub(q)](n)] and K[GL-[sub(q)](n)] -- 11. g-Schur Algebras -- 11.1. Polynomial representations of G[sub(q)] -- 11.2. The g-Schur algebra S[sub(q)](n,r).

11.3. S[sub(q)](n,r) as an endomorphism algebra -- 11.4. On the complete reducibility of G[sub(q)]-modules -- 11.5. S[sub(q)](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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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.