Actor of a crossed module of Leibniz algebras

José Manuel Casas, Rafael Fernández-Casado, Xabier García-Martínez, Emzar Khmaladze

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.

Keywords: Leibniz algebra, crossed module, representation, actor

2010 MSC: 17A30, 17A32, 18A05, 18D05

Theory and Applications of Categories, Vol. 33, 2018, No. 2, pp 23-42.

Published 2018-01-04.

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