Weak representability of actions of non-associative algebras

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 are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called strong amalgam. Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object $\mathcal{E}(X)$ for the actions on (or split extensions of) an object $X$. We actually obtain a partial algebra $\mathcal{E}(X)$, which we call external weak actor of $X$, together with a monomorphism of functors $\mathrm{SplExt}(-,X)\rightarrowtail \mathrm{Hom}(U(-),\mathcal{E}(X))$, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor $USGA(X)$ and that of $\mathcal{E}(X)$ are described in detail. We end with some open questions. This version is preceded by a corrigendum. Theorem 2.11 of the published article, which states that the variety of commutative associative algebras is weakly action representable, is false: we exhibit a $16$-dimensional algebra $X$ with zero multiplication for which the functor $\mathrm{SplExt}(-,X)$ admits no weak representation. Hence this variety is action accessible but not weakly action representable. The corrigendum amends the statements affected by this error, corrects Proposition 2.8, whose "if" part needs a strong amalgam, updates the definition of partial algebra, and lists some further minor corrections.

arXiv commentv4: 8-page corrigendum prepended to the published version. Theorem 2.11 (weak action representability of commutative associative algebras) is false; a counterexample is given and the affected statements are amended. The remaining results are unaffected, apart from minor corrections listed at the end

References in corpus (10)

• A characterisation of Lie algebras amongst anti-commutative algebras
• A characterisation of Lie algebras via algebraic exponentiation
• Algebras with representable representations
• Two-step nilpotent Leibniz algebras
• Derivations of two-step nilpotent algebras
• Biderivations of low-dimensional Leibniz algebras
• On the representability of actions of Leibniz algebras and Poisson algebras
• Isotopisms of nilpotent Leibniz algebras and Lie racks
• Associativity and the cosmash product in operadic varieties of algebras
• A characterisation of Lie algebras using ideals and subalgebras

Cited by in corpus (2)

• Action accessible and weakly action representable varieties of algebras
• On Lie-holomorphs of Leibniz algebras