Quasi-Centroids and Quasi-Derivations of Low Dimensional Associative Algebras

TL;DR

This paper investigates the properties of quasi-derivation and quasi-centroid algebras in low-dimensional associative algebras, providing classifications and explicit computations for dimensions two to four.

Contribution

It introduces the concepts of quasi-derivations and quasi-centroids in associative algebras and classifies these structures for low-dimensional cases.

Findings

Computed quasi-derivation algebras for all 2-4 dimensional associative algebras.

Computed quasi-centroid algebras for all 2-4 dimensional associative algebras.

Determined the dimensions of these algebras for each classified case.

Abstract

In this paper, we present some basic properties concerning the quasi-derivation algebra $QDer(\mathcal{A})$ and the quasi-centroid algebra $QC(\mathcal{A})$ of associative algebra $\mathcal{A}$. Furthermore, using the result on classification of two, three and four dimensional associative algebra, we compute, for all two, three and four dimensional associative algebras, quasi-derivation algebra $QDer(\mathcal{A})$ and the quasi-centroid algebra $QC(\mathcal{A})$ algebras and give their corresponding dimension.