Actor of a crossed module of Leibniz algebras

TL;DR

This paper extends the concept of biderivations to crossed modules of Leibniz algebras, constructing an actor object and exploring actions, kernels, centers, and inner/outer biderivations within this framework.

Contribution

It introduces a new notion of biderivation for crossed modules of Leibniz algebras and constructs the associated actor object, expanding the algebraic theory.

Findings

Construction of the actor object under certain conditions

Description of actions via equations in the category

Identification of the kernel with the center under specific conditions

Abstract

We extend to the category of crossed modules of Leibniz algebras the notion of biderivation via the action of a Leibniz algebra. This results into a pair of Leibniz algebras which allow us to construct an object which is the actor under certain circumstances. Additionally, we give a description of an action in the category of crossed modules of Leibniz algebras in terms of equations. Finally, we check that, under the aforementioned conditions, the kernel of the canonical map from a crossed module to its actor coincides with the center and we introduce the notions of crossed module of inner and outer biderivations.