Quantum Linear Groups

This volume begins with a general discussion of the theory of quantum groups. The authors view the theory as a natural extension of the theory of affine group schemes. They establish a number of foundational results, including the theory of induced representations and spectral sequences for quantum group cohomology. They then apply these results to give a detailed study on the quantum general linear group and its representation theory. Some of the central topics included are a development of quantum determinants, Frobenius kernals and their representation theory, high weight theory, and the generalization of various important theorems concerning the cohomology of vector bundles on the flag manifold. Finally, the authors use the theory to give a treatment of q-Schur algebras, proving, for example, that q-Schur algebras are quasi-hereditary.

Quantum Groups; Representation Theory of Quantum Groups; Quantum Matrix Spaces; Quantum Determinants; The Antipode and Quantum Linear Groups; Some Closed Subgroups; Frobenius Morphisms and Kernels; Global Representation Theory; Infinitesimal Representation Theory; The Generalization of Certain Important Theorems on the Cohomology of Vector Bundles on the Flag Manifold; q-Schur Algebras.