On the representability of actions of non-associative algebras

Abstract

It is well known that in the semi-abelian category Grp of groups, internal actions are represented by automorphisms. This means that the category Grp is action representable and the representing object, which is called the actor, is the group of automorphisms. 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 X being the Lie algebra of derivations Der(X). The notion of action representable category has proven to be quite restrictive: for instance, if a non-abelian variety V of 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, and we show that the existence of a weak representation is closely connected to the amalgamation property, which we use to prove that the variety CAssoc of commutative associative algebras is weakly action representable. Eventually, we give an application of the construction of the external weak actor in the context of varieties of unital algebras, which are ideally exact categories in the sense of G. Janelidze: 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 J. Brox, (Universidad de Valladolid, Spain), Alan S. Cigoli (Università degli Studi di Torino, Italy), Xabier García Martínez (Universidade de Vigo, Spain), Giuseppe Metere (Università degli Studi di Milano, Italy), Tim Van der Linden and Corentin Vienne (Université catholique de Louvain, Belgium).