Representations and cohomology for Frobenius-Lusztig kernels

الأصل · EN

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 has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.