Compatible Pairs of Low-Dimensional Associative Algebras and Their Invariants

TL;DR

This paper classifies compatible pairs of low-dimensional associative algebras over complex numbers and analyzes their algebraic invariants, providing a comprehensive understanding of their structure and symmetries.

Contribution

It offers the first systematic classification of compatible associative algebra pairs of dimension less than four and computes their key invariants.

Findings

Complete classification of compatible pairs of associative algebras under 4 dimensions.

Explicit computation of derivations, automorphisms, and other invariants for each class.

Identification of structural properties and symmetries of these algebraic pairs.

Abstract

A compatible associative algebra is a vector space endowed with two associative multiplication operations that satisfy a natural compatibility condition. In this paper, we investigate and classify compatible pairs of associative algebras of complex dimension less than four. Alongside these classifications, we systematically compute and analyze various algebraic invariants associated with them, including derivations, centroids, automorphism groups, quasi-centroids, Rota-Baxter operators, Nijenhuis operators, averaging operators, Reynolds operators, quasi-derivations, and generalized derivations.