On the representability of actions of algebras with several bilinear multiplications

Abstract

We study the representability of actions in varieties of algebras with $n \geq 1$ not necessarily associative bilinear multiplications. We first prove that, for homogeneous varieties, action accessibility is equivalent to being an Orzech category of interest. For an action accessible, operadic variety we then construct the external weak actor $\mathcal{E}(X)$ of an algebra $X$, an algebra with $n$ bilinear partial operations together with a natural monomorphism encoding the actions on $X$. We characterise the algebras admitting an actor in terms of $\mathcal{E}(X)$, and we deduce an obstruction to action representability: if, for some algebra $X$ with trivial multiplications, $\mathcal{E}(X)$ does not satisfy a multilinear identity of the variety, then the variety is not action representable. Computing the external weak actor explicitly, we obtain that associative and matching dialgebras are weakly action representable but not action representable, and that Poisson algebras, left-right non-commutative Poisson algebras and matching Lie algebras are not action representable. Finally, we introduce the reduced type $(p,q)$ of such a variety and prove that, over a field of characteristic different from $2$, an operadic variety of reduced type $(p,0)$, $(0,q)$ or $(1,1)$ is action representable if and only if it is term equivalent to the variety of Lie algebras or to that of abelian algebras. For the reduced types $(p,0)$ and $(0,q)$, the arguments require no assumption on the cardinality of the field, whereas the case $(1,1)$ requires the field to be infinite.