On Derivations of Tensor Products of Perm Algebras and Associative Algebras
José Gregorio Rodríguez-Nieto, Olga Patricia Salazar-Díaz, Andrés Sarrazola-Alzate, Raúl Velásquez
TL;DR
The paper addresses the problem of describing derivations and diderivations on tensor product dialgebras formed from a perm algebra and a unital associative algebra. It develops a decomposition framework that expresses any derivation as a combination of derivations from each factor together with interaction maps, and it extends this to diderivations with a dual role on the perm component and the associative component; explicit coordinate formulas are provided with respect to natural bases. Key contributions include a concrete decomposition theorem and coordinate expressions that generalize classical results beyond purely associative settings to dialgebras and perm algebras, enabling systematic analysis of infinitesimal symmetries in these tensor product structures. The results offer computational tools and theoretical insight, and point to operadic generalizations and cohomological/deformation perspectives as promising directions for further work.
Abstract
The study of derivations and their generalizations on non-associative algebras has proven to be fundamental in understanding the internal symmetries and algebraic dynamics of such structures. In this paper, we investigate derivations and diderivations of tensor product dialgebras arising from the combination of a perm algebra and a unital associative algebra. We provide decomposition theorems that characterize these operators in terms of derivations of the individual factors and suitable multiplication maps. Explicit coordinate formulas are also derived, allowing concrete descriptions of the action of derivations and diderivations with respect to natural bases. These results extend classical decomposition theorems for tensor products beyond the associative setting, highlighting the interplay between perm algebras and non-associative algebraic frameworks.