Cohomology for infinitesimal unipotent algebraic and quantum groups

TL;DR

This paper investigates the cohomology structures of Frobenius kernels in unipotent and parabolic algebraic groups and their quantum analogs, providing explicit ring structures and module computations.

Contribution

It determines the cohomology ring structure of Frobenius kernels for unipotent radicals and extends results to quantum groups, including module computations.

Findings

Cohomology ring structure of (U_J)_1 explicitly determined

New results on H^•((P_J)_1, L(λ)) as an L_J-module

Generalizations to quantum groups achieved

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 $H^\bullet((P_J)_1,L(\lambda))$ as an $L_J$-module where $L(\lambda)$ is a simple $G$-module with high weight $\lambda$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situation.