In this work quantum groups are thought of as a generalization of affine algebraic groups. The generalization consists in allowing the coordinate ring to be noncommutative.
In this set-up the authors develop quite generally several foundational results in the representation theory for quantum groups - in particular on induction and cohomology. Then they study in detail the quantum linear groups. Here they obtain generalizations of many basic theorems from the representation theory of reductive algebraic groups. They also apply this theory to prove new results about q-Schur algebras.
In a parallel development the reviewer, P. Polo and Wen Kexin have obtained similar results for quantized enveloping algebras [Invent. Math. 104, 1-59 (1991; see the following review)]. (Note that by the results of P. Polo in the appendix of loc. cit. it is possible to translate directly from the quantized enveloping algebra of \({\mathfrak g}{\mathfrak l}_ n\) to the quantum linear group.)