On certain Hochschild cohomology groups for the small quantum group

TL;DR

This paper computes Hochschild cohomology groups for blocks of the small quantum group using sheaf cohomology methods, confirming conjectures and revealing module structures in specific cases.

Contribution

It applies a sheaf cohomology BGG method to compute Hochschild cohomology of small quantum group blocks, verifying conjectures and analyzing module structures.

Findings

Computed Hochschild cohomology groups for various blocks.

Confirmed the conjectures of Lachowska-Qi regarding the center.

Determined the module structure for the nontrivial singular block of sl_3.

Abstract

We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for $\mathfrak{g} = \mathfrak{sl}_3$, we obtain the H*$(\mathfrak{u}, \mathbb{C}) = \mathbb{C}[\mathcal{N}]$-module structure of HH*$(\mathfrak{u}_{\lambda})$.