Weak representability of actions: a study from groups to non-associative algebras

It is well known that in the semi-abelian category Grp of groups, internal actions are represented by authomorphisms. This means that the category Grp is action representable and the representing object, which is called the actor, is the group of authomorphisms. Another example of action representable category is the variety Lie of Lie algebras over a fixed field F, with the actor of a Lie algebra g being the Lie algebra of derivations Der(g). The notion of action representable category has proven to be quite restrictive: for instance, if a non-abelian variety Vof non-associative algebras over an infinite field F is action representable, then V= Lie. More recently G. Janelidze introduced the notion of weakly action representable category, which includes a wider class of categories, such as the variety Assoc of associative algebras and the variety Leib of Leibniz algebras. In this talk we show that for an algebraically coherent and operadic variety V and an object X of V, it is always possible to construct a partial algebra E(X), called external weak actor of X, and a natural monomorphism of functors Act(−,X) >--> Hom(U(−),E(X)), where U:V-->PAlg denotes the forgetful functor. Moreover, for any other object B of V, we provide a complete description of the homomorphisms of partial algebras which identify the actions of B on X in V. Eventually, we give an application of this construction in the context of varieties of unital algebras: we prove that, if V= Alt is the variety of alternative algebras and X is a unital alternative algebra, then E(X)=X is the actor of X. In other words, unital alternative algebras, such as the algebra O of octonions, have representable actions. This is joint work with Alan S. Cigoli (Università degli Studi di Torino, Italy), Xabier García Martínez (Universidade de Vigo, Spain), Giuseppe Metere (Università degli Studi di Palermo, Italy), Tim Van der Linden and Corentin Vienne (Université catholique de Louvain, Belgium).