$σ$-matching and interchangeable structures on certain associative algebras
Mykola Khrypchenko
TL;DR
This work develops a unified framework for σ-matching, interchangeable, and totally compatible pairs of binary operations on associative algebras, linking these notions to Hochschild 2-cocycles and centroid/center actions. It provides explicit classifications across several algebra families: unital algebras (where mutations by central elements govern the structures), idempotent algebras (with concrete forms in rectangular-band semigroup algebras and algebras with enough idempotents), and free non-unital (commutative) algebras (where all three notions coincide with mutations driven by the centroid, often scalar in the free cases). Key results include explicit forms such as $e_{ij}*e_{kl} = \\lambda e_{il}$ in rectangular bands, and general descriptions in terms of fixed elements of the centroid or the ambient matrix-like space $M$, clarifying when σ-matching implies total compatibility. The findings illuminate when different compatibility notions coincide or diverge, and provide concrete algebraic constructions that tie into representation theory and operad-theoretic perspectives on compatible products.
Abstract
We describe $σ$-matching, interchangeable and, as a consequence, totally compatible products on some classes of associative algebras, including unital algebras, the semigroup algebras of rectangular bands, algebras with enough idempotents, free non-unital associative algebras and free non-unital commutative associative algebras.