Cohomology for infinitesimal unipotent algebraic and quantum groups
math.RT
/ Authors
/ Abstract
In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and its unipotent radical $U_J$, we determine the ring structure of the cohomology ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing $H^\bullet((P_J)_1,L(λ))$ as an $L_J$-module where $L(λ)$ is a simple $G$-module with high weight $λ$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situation.