A Natural Extension of the Universal Enveloping Algebra Functor to Crossed Modules of Leibniz Algebras

Abstract

Abstract. The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category LM in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.

Braiding for categorical algebras and crossed modules of algebras II: Leibniz algebras

“…For this purpose, we give the following propositions of which we omit their proofs because they are immediate. The first is symmetrical to the construction we can see in [4] for crossed modules with right actions. by m,…”

“…The functor Ψ on objects is described in the following proposition. In [4], we can see that the previous construction can be extended to crossed modules of Lie algebras in LM K . They did a crossed module with a right action.…”

“…In this section, we will use the idea of Loday and Pirashvili ([11]) to see the Leibniz K-algebras as a particular case of a Lie algebra in the monoidal category of linear maps LM K , also known as the Loday-Pirashvili category ( [4,12]). Using this, we will try to define the concept of braiding in the case of Leibniz algebras taking advantage of the fact that they will be a particular case of braidings for the corresponding ideas over Lie objects in that category.…”

Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras

“…, is a Leibniz K-algebra. In [2] we can see that the previous construction can be extended to crossed modules of Lie algebras in LM K . They did a crossed module with a right action.…”

“…In this section we will use the idea of Loday and Pirashvili ( [7]) that allows us to see the Leibniz K-algebras as a particular case of a Lie algebra in the monoidal category of linear maps LM K , also known as the Loday-Pirashvili category ( [2,8]). Using this, we will try to define the concept of braiding in the case of Leibniz algebras taking advantage of the fact that they will be a particular case of braidings for the corresponding ideas over Lie objects in that category.…”