Compatible Pairs of Low-Dimensional Associative Algebras and Their Invariants
Ahmed Zahari Abdou Damdji, Bouzid Mosbahi
TL;DR
The paper addresses the classification of complex compatible associative algebras with dimension $<4$ and the computation of a broad suite of invariants. It derives and solves the compatibility equation $(a mu1 b) mu2 c + (a mu2 b) mu1 c = a mu1 (b mu2 c) + a mu2 (b mu1 c)$ for two multiplications, producing explicit 2D–4D families and determining automorphism groups, with invariant calculations performed via Maple/Mathematica. It contributes complete low-dimensional classifications, explicit algebra families, and comprehensive invariant analyses including derivations, centroids, and various linear operators (Rota-Baxter, Nijenhuis, averaging, Reynolds, quasi-derivations, generalized derivations), all accompanied by matrix characterizations for the 2D–4D cases. The results supply concrete examples and a computational framework that can inform deformation theory, operad theory, and noncommutative geometry.
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.