On the representability of actions of Leibniz algebras and Poisson algebras

TL;DR

This paper investigates the conditions under which actions of Leibniz and Poisson algebras can be represented, introducing a universal actor for Poisson algebra actions and establishing the weak action representability of Leibniz algebras.

Contribution

It proves that Leibniz algebras are weakly action representable and explicitly constructs a universal strict general actor for Poisson algebra actions.

Findings

Leibniz algebras are weakly action representable.

A universal strict general actor for Poisson algebra actions is explicitly described.

Characterization of acting morphisms in Leibniz algebras.

Abstract

In a recent paper, motivated by the study of central extensions of associative algebras, G. Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras by describing explicitly a universal strict general actor.