On the additivity of n-multiplicative isomorphisms, derivations, and related maps in axial algebras
Daniel Eiti Nishida Kawai, Henrique Guzzo, Bruno Leonardo Macedo Ferreira
TL;DR
The paper addresses when $n$-multiplicative isomorphisms, $n$-multiplicative derivations, elementary maps, and Jordan elementary maps on axial algebras are additive under Martindale-type conditions. It introduces nullifying functions as a unifying tool to reduce additivity questions to checking a single function vanishes. It proves general additivity results for $\mathcal{J}(\alpha)$- and $\mathcal{M}(\alpha,\beta)$-algebras, including the open $\beta=\frac{1}{2}$ case, by establishing Martindale-like conditions under which all nullifying functions vanish. The work extends classical preservers results to non-associative axial algebras, with explicit examples such as Jordan algebras, Norton–Sakuma-type algebras, and Highwater showing the applicability.
Abstract
In this paper, we demonstrate that several classes of functions, specifically n-multiplicative isomorphisms, derivations, elementary maps, and Jordan elementary maps on a class of algebras that includes Jordan algebras with idempotents, J(a)-axial algebras and M(a,b)-axial algebras, are additive under appropriate conditions, which may be referred to as Martindale-type conditions. Furthermore, we answer the question left open in the recent article titled "Multiplicative isomorphisms and derivations on axial algebra."