Representations and cohomology for Frobenius–Lusztig kernels

Let Uζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ℓ-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras Uζ(Gr) of Uζ, called Frobenius–Lusztig kernels, which generalize the Frobenius kernels Gr of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible Uζ(Gr)-modules and discuss their characters. We then study the cohomology rings for the Frobenius–Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius–Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.