# Structure and classification of Hom-associative algebras

## TL;DR

This paper explores the structure, classification, and deformation of Hom-associative algebras, including simple cases, derivations, cohomology, and irreducible components, advancing understanding of their algebraic properties.

## Contribution

It provides a classification of low-dimensional Hom-associative algebras, characterizes simple cases, and analyzes their derivations and cohomology groups.

## Key findings

- Classification of n-dimensional Hom-associative algebras for n ≤ 3
- Examples of deformations of 2x2 matrix algebra into simple Hom-associative algebras
- Computation of Hom-Type Hochschild cohomology groups

## Abstract

The purpose of this paper is to study the structure and the algebraic varieties of Hom-associative algebras. We give characterize multiplicative simple Hom-associative algebras and show some examples deforming the $2\times 2$-matrix algebra to simple Hom-associative algebras. We provide a classification of $n$-dimensional Hom-associative algebras for $n\leq3$. Then study their derivations and compute small Hom-Type Hochschild cohomology groups. Furthermore, we discuss their irreducible components.