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.
Cite
@article{arxiv.2405.18243,
title = {Compatible Associative Algebras and Some Invariants},
author = {Erik Mainellis and Bouzid Mosbahi and Ahmed Zahari},
journal= {arXiv preprint arXiv:2405.18243},
year = {2024}
}
Comments
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