Regular representations of the quantum groups at roots of unity

TL;DR

This paper investigates the bimodule structure of quantum function algebras at roots of unity, revealing a filtration related to Weyl modules and computing Hochschild cohomology.

Contribution

It introduces a new filtration structure for quantum function algebras at roots of unity and computes their Hochschild cohomology, advancing understanding of their algebraic properties.

Findings

Filtration of the bimodule structure with factors isomorphic to tensor products of dual Weyl modules

Explicit computation of the 0-th Hochschild cohomology of the algebra at roots of unity

Enhanced understanding of quantum groups at roots of unity

Abstract

We study the bimodule structure of the quantum function algebra at roots of 1 and prove that it admits an increasing filtration with factors isomorphic to the tensor products of the dual of Weyl modules $V_\lambda^* \otimes V_{- \omega_0 \lambda}^*$. As an application we compute the 0-th Hochschild cohomology of the function algebra at roots of 1.