Representations and cohomology for Frobenius-Lusztig kernels

Let $U_\zeta$ be the quantum group (Lusztig form) associated to the simple Lie algebra $\mathfrak{g}$, with parameter $\zeta$ specialized to an $\ell$-th root of unity in a field of characteristic $p>0$. In this paper we study certain finite-dimensional normal Hopf subalgebras $U_\zeta(G_r)$ of $U_\zeta$, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels $G_r$ 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 $U_\zeta(G_r)$-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.Comment: 26 pages. Incorrect references fixe

On injective modules and support varieties for the small quantum group

Let $u_\zeta(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $\zeta$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_\zeta(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_\zeta(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_\zeta(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U_\zeta(G_r)$ of the quantized enveloping algebra $U_\zeta(g)$. Now suppose that $M$ lifts to a $U_\zeta(g)$-module. Using a new rank variety type result for the support varieties of $u_\zeta(g)$, we prove that the injectivity of $M$ for $u_\zeta(g)$ can be detected by the restriction of $M$ to a single root subalgebra.Comment: 21 pages. Title changed from previous version. Various other minor corrections and changes mad

Corrigendum to "On injective modules and support varieties for the small quantum group"

The proof of Theorem 5.12 in [C.M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. 2011 (2011), 2263-2294] does not make sense as written because the algebra $u_\zeta(\mathfrak{b}_\alpha^+)$ need not be a Hopf subalgebra of $u_\zeta(\mathfrak{b}^+)$ unless $\alpha$ is a simple root. This note describes how the proof should be modified to work around this fact.Comment: 2 pages; Corrigendum to [C.M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. 2011 (2011), 2263-2294