# Derivations and Hochschild cohomology of zigzag algebras

## TL;DR

This paper investigates the structure of zigzag algebras derived from certain graphs, proving that all Jordan derivations are derivations and determining the dimension of their Hochschild cohomology groups.

## Contribution

It establishes that all Jordan derivations of zigzag algebras are derivations and computes the Hochschild cohomology dimension, extending results to derived equivalent algebras.

## Key findings

- All Jordan derivations are derivations for zigzag algebras.
- The dimension of the first Hochschild cohomology group is one.
- The result extends to algebras derived equivalent to zigzag algebras.

## Abstract

Let $\Gamma$ be a connected graph without loops, cycles or multiple edges and $Z(\Gamma)$ the corresponding zigzag algebra. Then every Jordan derivation of $Z(\Gamma)$ is a derivation. Moreover, we will prove that the dimension of 1th Hochschild cohomology group of $Z(\Gamma)$ is one by computing the dimensions of linear spaces spanned by derivations and inner derivations. This implies that the dimension of the 1th Hochschild cohomology group of each algebra derived equivalent to a zigzag algebra is 1.