Hochschild cohomology of $q$-Schur algebras

TL;DR

This paper calculates the Hochschild cohomology of blocks of $q$-Schur algebras, revealing structural properties and establishing algebraic isomorphisms through graded surjections and derived equivalences.

Contribution

It introduces methods to compute Hochschild cohomology of $q$-Schur algebras using graded surjections and derived equivalences, expanding understanding of their algebraic structure.

Findings

Constructed graded algebra surjections between Hochschild cohomologies of quasi-hereditary algebras.

Established graded algebra isomorphism of Hochschild cohomologies via derived equivalence.

Focused on the even part of the Hochschild cohomology ring.

Abstract

We compute the Hochschild cohomology of any block of $q$-Schur algebras. We focus the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of $q$-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all $q$-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derived equivalence.