Action representability in categories of unitary algebras

Abstract: In a recent article [13], G. Janelidze introduced the concept of ideally exact categories as a generalization of semi-abelian categories, aiming to incorporate relevant examples of non-pointed categories, such as the categories $\textbf{Ring}$ and $\textbf{CRing}$ of unitary (commutative) rings. He also extended the notion of action representability to this broader framework, proving that both $\textbf{Ring}$ and $\textbf{CRing}$ are action representable. This article investigates the representability of actions of unitary non-associative algebras. After providing a detailed description of the monadic adjunction associated with any category of unitary algebra, we use the construction of the external weak actor [4] in order to prove that the categories of unitary (commutative) associative algebras and that of unitary alternative algebras are action representable. The result is then extended for unitary (commutative) Poisson algebras, where the explicit construction of the universal strict general actor is employed.