Classification of some operators on compatible ternary Leibniz algebras
Kol Béatrice Gamou, Ahmed Zahari Abdou, Ibrahima Bakayoko
TL;DR
The paper develops a cohesive framework for operator theory on (compatible) ternary Leibniz algebras, connecting binary Leibniz structures to their ternary counterparts via the construction $T(L)$. It delivers explicit classifications of small-dimensional (2D and 3D) ternary Leibniz algebras and compatible cases, and provides exhaustive descriptions of derivations, averaging operators, Rota-Baxter and Reynolds operators, Nijenhuis operators, and centroids on these algebras. The results include detailed matrix or parametrized forms for each operator across all classified algebras, enabling precise structural analysis and potential applications in nonassociative algebra and mathematical physics. By establishing how these operators behave and intertwine with the ternary structure, the work offers a foundation for further exploration of compatible ternary Leibniz systems and their algebraic deformations. The combination of classification and operator descriptions enhances both theoretical insight and computational accessibility for small-dimensional examples.
Abstract
In this paper, we establish some basic properties of certain operators (element of centroids, averaging operators, derivations, Nijenhuis operators, Rota-Baxter operators) on (compatible) ternary Leibniz algebras and give the classification of ternary Leibniz algebras, classification of compatible ternary Leibniz algebras. Then, we give the descriptions of operators on found elements of these classifications.