Hochschild cohomology groups of 5-dimensional complex nilpotent associative algebras

Abstract

This paper explores the structure of low-dimensional cohomology groups in the context of complex nilpotent associative algebras. Specifically, we study 5-dimensional complex nilpotent associative algebras satisfying $\mathcal{A}^4 = 0$ and $\mathcal{A}^3 \neq 0$. Using their isomorphism invariants, we compute and present the zeroth and first Hochschild cohomology groups, $H^0(\mathcal{A}, \mathcal{A})$ and $H^1(\mathcal{A}, \mathcal{A})$, in explicit matrix form. These results show how cohomology helps to identify and classify different associative algebras.