Weak representability of actions of non-associative algebras

TL;DR

This work investigates weak representability of internal actions (WRA) in varieties of non-associative algebras over a field, situating the study in semi-abelian categories. It classifies action-accessible, operadic quadratic varieties with degree-2 identities and proves weak action representability for several natural families, linking WRA to amalgamation properties. The paper introduces an external weak actor E(X) and relates it to the universal strict general actor USGA(X), providing a concrete quadratic-case framework to compute E(X) and identify when it acts as a genuine actor across various subvarieties such as CA ssoc, JJord, and Nil2 variants, with explicit identifications in several classical settings. It closes with open questions on subvarieties, initial representations, and unitary-closed extensions, outlining directions for further algorithmic construction of weak representations in broader contexts.

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

Theorems & Definitions (64)

• Definition 1.1: BJK2
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.6
• Lemma 1.7
• Remark 1.8
• Theorem 1.9: GM-VdL2GM-VdL3
• Theorem 1.10: Tim
• Proposition 1.11