Weak representability of actions for categories of Leibniz algebras and Poisson algebras

Abstract

It is well known that, in the semi-abelian category LieAlg of Lie algebras over a field F algebra actions are represented by derivations. From a categorical point of view, this means that the category LieAlg is action representable and the representing object, which is called the actor, is the Lie algebra of derivations. The notion of action representable category has proven to be quite restrictive. For example, if a variety V of non-associative algebras over F is action representable, then V must be the category LieAlg. More recently G. Janelidze introduced the notion of weakly action representable category, which includes a wider class of categories. In this talk we explain that the category LeibAlg of (right) Leibniz algebras and the category Pois of (non-commutative) Poisson algebras are weakly action representable. In both cases we give we give construction of the weak actor [X] of a fixed object X and, given two objects X,B, we provide a complete description of the acting morphisms B-->[X], i.e. of the morphisms which identify the split extensions of B by X. This is joint work with Alan Cigoli (University of Turin) and Giuseppe Metere (University of Palermo).