On weak action representability, action accessibility and varieties of non-associative algebras

It is well known that in the semi-abelian category of groups, internal actions are represented by automorphisms. This means that the category of groups is action representable with the actor of a group being its group of automorphisms. The notion of action representability has proven to be quite restrictive: for instance, it was shown that the only non-abelian variety of non-associative algebras which is action representable is the variety of Lie algebras. More recently G. Janelidze introduced the concept of weakly action representable category, which includes a wider class of categories, such as the variety of associative algebras and the variety of Leibniz algebras. This notion was studied in the context of varieties of algebras: it was shown that every object X of an algebraically coherent variety of non-associative algebras admits an external weak representation, which consists of a partial algebra E(X) together with a monomorphism of functors Act(-,X) >--> Hom(U(-),E(X)), where U denotes the forgetful functor. The aim of this talk is to investigate the relationship between action accessibility and weak action representability in the frame of varieties of non-associative algebras. Using an argument of J. R. A. Gray in the setting of groups, we prove that the varieties of k-nilpotent (k>2) and n-solvable (n>1) Lie algebras are not weakly action representable. These are the first known examples of action accessible varieties of non-associative algebras that fail to be weakly action representable, establishing that a subvariety of a (weakly) action representable variety of non-associative algebras need not be weakly action representable. We then aim to study the representability of actions in the context of categories of unitary non-associative algebras, which are ideally exact in the sense of G. Janelidze. After describing the monadic adjunction associated with any category of unitary algebras, we prove that the categories of unitary associative algebras, unitary alternative algebras and unitary Poisson algebras are action representable. This is joint work with Xabier García Martínez (Universidade de Santiago de Compostela, Spain) and Federica Piazza (Università degli Studi di Messina, Italy).