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 $E\left(X\right)$ for the actions on (or split extensions of) an object X. We actually obtain a partial algebra $E\left(X\right)$, which we call external weak actor of X, together with a monomorphism of functors $\mathrm{SplExt}(-,X)\rightarrowtail \mathrm{Hom}\left(U\right(-),E(X\left)\right)$, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor $\mathrm{USGA}\left(X\right)$ and that of $E\left(X\right)$ are described in detail. We end with some open questions.