Hochschild cohomology groups of 5-dimensional complex nilpotent associative algebras

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 ${\mathcal{A}}^3\mathrm{\ne }0$. 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.