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 $\mathrm{SplExt}(-,X)\rightarrowtail \mathrm{Hom}(U(-),E(X))$, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor $\mathrm{USGA}(X)$ and that of $E(X)$ are described in detail. We end with some open questions.

Final version, accepted for publication

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Cited by in corpus (2)

• Action accessible and weakly action representable varieties of algebras
• On Lie-holomorphs of Leibniz algebras