On the representability of actions of Leibniz algebras and Poisson algebras
Alan S. Cigoli, Manuel Mancini, Giuseppe Metere
TL;DR
The paper investigates action representability in semi-abelian contexts, focusing on Leibniz and Poisson algebras. It shows that Leibniz algebras form a weakly action representable category with the weak actor given by the biderivation algebra Bider(h), and characterizes acting morphisms via a derived-action condition. For Poisson algebras, it constructs a universal strict general actor [V] in the ambient category of algebras with two bilinear operations, establishing a natural embedding of split extensions into Hom(U(-),[V]); however, [V] is not generally Poisson, so PoisAlg is not in general weakly action representable. The paper also develops [V]_c for commutative Poisson algebras, providing a similar universal actor in CPoisAlg and highlighting special cases where a natural isomorphism holds. Open problems remain about the weak action representability of PoisAlg and CPoisAlg in full generality, but the USGA framework yields concrete tools for understanding actions and extensions in these categories.
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.