# Hochschild cohomology of $m$-Cluster tilted algebras of type $\widetilde{\mathbb{A}}$

## TL;DR

This paper calculates the Hochschild cohomology dimensions of $m$-cluster tilted algebras of type $ ilde{ ext{A}}$, explores conditions for non-trivial algebraic structures, and discusses limitations in classifying these algebras via cohomology.

## Contribution

It provides explicit Hochschild cohomology dimensions for these algebras and identifies conditions for non-trivial multiplicative structures, highlighting classification limitations.

## Key findings

- Hochschild cohomology dimensions are explicitly computed.
- Conditions for non-trivial Gerstenhaber algebra structures are established.
- Hochschild cohomology does not fully determine the derived class of gentle $m$-cluster tilted algebras.

## Abstract

In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\tilde{\mathbb{A}}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $\Lambda$ of type $\tilde{\mathbb{A}}$ such that the Gerstenhaber algebra $\operatorname{HH}^*(\Lambda)$ has non-trivial multiplicative structures. We also show that the derived class of gentle $m$-cluster tilted algebras is not always completely determined by the Hochschild cohomology.