On the representability of actions of non-associative algebras

We study the categorical-algebraic condition of internal actions being weakly representable in the context of non-associative algebras over a field. It is known that such varieties are action accessible if and only if they are Orzech categories of interest and it is also known that both these conditions are implied by weakly representable actions in this context.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 (so commutative or anti-commutative algebras) and to study the representability of actions for them. Moreover, we prove that the varieties of two-step nilpotent commutative and anti-commutative algebras are weakly action representable.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 of a variety of non-associative algebras V. We actually obtain a partial algebra E(X), which we call external weak actor of X, together with a natural monomorphism of functors SplExt(-,X) >--> Hom_PAlg(U(-),E(X)), where PAlg is the category of partial algebras and U: V --> PAlg denotes the forgetful functor, which we study in detail in the case of Leibniz algebras, where E(X) = Bider(X) is the Leibniz algebra of biderivations of X. Furthermore, the relations between the construction of the universal strict general actor USGA(X) and that of E(X) are thoroughly described.Eventually, we study the representability of actions of the category of (non-commutative) Poisson algebras, showing a possible direction for the construction of the external weak actor for any action accessible variety of algebras with two non-necessarily associative bilinear operations. We conclude with some open problems.