Compatible Associative Algebras and Some Invariants

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.

The paper rests on an erroneous conception of classifying compatible structures. In particular, isomorphism classes of compatible algebras cannot be adequately described via nice pairs of the underlying algebras. A proper classification must consider the entire structure and how the operations interact, and should take the form of listing nonzero multiplications on basis elements