Induced Representations of Hopf Algebras: Applications to Quantum Groups at Roots of 1

Introduction This paper serves three purposes. The first is to build up a connection between the representation theories of the quantum enveloping algebra U of a semisimple Lie algebra and the quantum groups defined in [15]. The second one is to study the structure of the derived functors of the induction functor, defined in [1]. At each integral weight, these derived functors give rise to certain U-modules resembling the cohomology groups of line bundles over the flag varieties of the corresponding algebraic groups. Finally we would like to generalize the representation theory of the hyperalgebras of algebraic groups to Hopf algebras. Since Lusztig first studied the quantum deformations of certain simple representations of semisimple Lie algebras in [10], he has established certain connections between the representations of U and the representations of an algebraic group in prime characteristic in [11, 13, 14]. It has been a hope that some of the problems in modular representation theory of algebraic groups could be solved through the representations of quantum enveloping algebras. The representation theory of U resembling that of algebraic groups is studied in [1] by Andersen, Polo and Wen (APW for short). Following another line of defining quantum groups, Parshall and Wang [15] studied the representation theory of quantum groups in terms of comodules of the quantum coordinate algebra. They have studied the linear quantum groups in detail and many of the classical theories have been extended in this special case. Out of curiosity, one would like to see how these two theories are related. In the appendix to [1], Polo uses the representation theory developed in that paper and shows that the coordinate algebra (defined by Lusztig in [14]) of U is isomorphic to the quantum coordinate algebra of SL q (n) (with v → q −1) if the Lie algebra has root system A n−1. In this paper we further establish a connection between the induction functors and the associated cohomology theories through the isomorphism of quantum coordinate algebras. This paper is organized as follows. In Section 1, we give a general setting on the coordinate algebra, which is called the Hopf dual, of a certain representation category,