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The universal enveloping algebra functor between Leibniz and associativealgebras defined by Loday and Pirashvili is extended to crossed modules. Weprove that the universal enveloping crossed module of algebras of a crossedmodule of Leibniz algebras is its natural generalization. Then we construct anisomorphism between the category of representations of a Leibniz crossed moduleand the category of left modules over its universal enveloping crossed moduleof algebras. Our approach is particularly interesting since the actor in thecategory of Leibniz crossed modules does not exist in general, so the techniqueused in the proof for the Lie case cannot be applied. Finally we move on to theframework of the Loday-Pirashvili category $\mathcal{LM}$ in order tocomprehend this universal enveloping crossed module in terms of the Lie crossedmodules case.

A natural extension of the universal enveloping algebra functor to crossed modules of Leibniz algebras