Free Reductive Lie Algebra Pairs of Lie-Yamaguti algebras

Abstract

The goal of this article is to show the categorical links between on the one hand the category of reductive Lie algebra pairs $\mathcal{R}\mathcal{L}\mathcal{P}$ and on the other hand the category of Lie-Yamaguti algebras $\mathcal{L}\mathcal{Y}$. The fact that the well-known construction of an enveloping algebra associating to a Lie-Yamaguti algebra a reductive Lie algebra pair is not functorial leads us to the main construction of the article, namely a left adjoint to the natural restriction functor $G:\mathcal{R}\mathcal{L}\mathcal{P}\to \mathcal{L}\mathcal{Y}$. As a final result we observe that the construction of the enveloping algebra becomes functorial when one restricts the morphisms of the categories $\mathcal{R}\mathcal{L}\mathcal{P}$ and $\mathcal{L}\mathcal{Y}$ to the surjective ones. Then it becomes a right adjoint to the restriction functor.