Compatible Associative Algebras and Some Invariants

TL;DR

This paper classifies low-dimensional compatible associative algebras and explores their invariants, including derivations, automorphisms, and Rota-Baxter operators, advancing understanding of their structural properties.

Contribution

It provides the first comprehensive classification of compatible associative algebras of dimension less than four and analyzes their key invariants.

Findings

Classified all compatible associative algebras of dimension < 4

Determined derivations, automorphisms, and centroids for these algebras

Characterized invariants like Rota-Baxter operators and second cohomology

Abstract

A compatible associative algebra is a vector space equipped with two associative multiplication structures that interact in a certain natural way. This article presents the classification of these algebras with dimension less than four, as well as the classifications of their corresponding derivations, centroids, automorphisms, and quasi-centroids. We then characterize a selection of further invariants such as Rota-Baxter operators and second cohomology for some specific examples.