Action representability in categories of unitary algebras
Manuel Mancini, Federica Piazza
TL;DR
This work extends the theory of action representability to categories of unitary algebras by leveraging external weak actors and the universal strict general actor. It proves action representability for unitary (commutative) associative algebras ($\mathbf{Assoc}_1$) and unitary alternative algebras ($\mathbf{Alt}_1$), with the actor identified as $X$ (via bimultipliers or related multiplier structures), and extends the result to unitary (commutative) Poisson algebras by showing the actor is the Poisson center $\mathrm{Z}(X)$ (and, in the commutative case, a streamlined actor). The paper also develops the framework for Poisson algebras using the universal strict general actor $[X]$, establishing representability in the Poisson setting and highlighting open questions about weak representability for Poisson algebras. Overall, it broadens the landscape of action representability beyond rings to a wider class of unitary, non-associative algebras, clarifying the structure of actors in these categories and linking to known results in idealized exactness theory.
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.