Crossed modules of ternary Leibniz algebras
Kol Béatrice Gamou, Ibrahima Bakayoko
TL;DR
The paper addresses constructing and relating crossed modules across triassociative, Leibniz, and ternary Leibniz algebras. It develops operator-based and categorical pipelines, notably via Rota-Baxter, Nijenhuis, and Reynolds operators, to lift binary crossed-module data to ternary Leibniz crossed modules using the functor $T(-)$ and semidirect products, with explicit compatibility rules. Key contributions include precise definitions and constructions of triassociative and Leibniz crossed modules, explicit ternary Leibniz crossed-module structures derived from these via $T(A)$ and related products, and the extension to Rota-Baxter-assisted actions that preserve morphisms. The work enriches the interplay between binary and ternary algebraic frameworks, offering a structured approach that could inform cohomology, categorification, and higher-homotopy perspectives in algebra.
Abstract
The aim of this paper is to construct triassociative algebras (from operators), new actions and crossed modules from a given one, and to make the connexion between these notions on Leibniz algebras or triassociative algebras and the corresponding notions on ternary Leibniz algebras.