Weak Representability of Actions of Non-Associative Algebras

Abstract

We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field. Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam. Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object E(X) for the actions on (or split extensions of) an object X. We actually obtain a partial algebra E(X), which we call external weak actor of X, together with a monomorphism of functors SplExt(-,X) >--> Hom(-,E(X)), which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor USGA(X) and that of E(X) are described in detail. We end with some open questions.