arXiv subjects

Corentin Vienne

Publications and source records attributed to Corentin Vienne.

5 recordsLinked to original sources

arXiv2025

Categorical-algebraic aspects of Heyting semilattices

This article gives an overview of some key categorical-algebraic properties of the variety of Heyting semilattices, with the aim of correcting a misconception in the literature. We confirm that the category of Heyting semilattices is not algebraically coherent, even though it satisfies a strong version of the so-called Smith is Huq condition (on the equivalence of two types of commutators). We also prove that Higgins commutators of normal subobjects are normal, as a consequence of the fact that Heyting semilattices form an arithmetical category. We provide an elementary characterisation of when a pair of subobjects commutes, and use this in the construction of two counterexamples. We further show that centralisers exist, centralisers of normal monomorphisms are normal monomorphisms, and normal monomorphisms are closed under composition. We study the latter condition in detail. On the other hand, we show that the category of Heyting semilattices does not satisfy normality of unions. Hence, it is not action accessible and so it does not admit all normalisers. In particular, this means that the known implication between action accessibility and the condition requiring the existence of centralisers of normal monomorphisms which are themselves normal, is strict.

math.CT↗

arXiv2025

On the representability of actions of unital algebras

Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the category of all unital algebras is not even action accessible. We then consider this problem in the context of operadic, action accessible, unit-closed varieties. Using the construction of the external weak actor, we prove that for any algebra $X$ in such a variety $\mathsf{V}$, the canonical map into its external weak actor is an isomorphism if and only if $X$ is unital. Consequently, the ideally exact category $\mathsf{V}_1$ of unital algebras in $\mathsf{V}$ is action representable, and the actor of $X$ is $X$ itself. Finally, we prove action representability for unital Poisson algebras via an explicit construction of the universal strict general actor.

math.CT↗

arXiv2023

Weak representability of actions of non-associative algebras

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.

math.CT↗

arXiv2022

Associativity and the cosmash product in operadic varieties of algebras

In this article, we characterise the operadic variety of commutative associative algebras over a field via a (categorical) condition: the associativity of the so-called cosmash product. This condition, which is closely related to commutator theory, is quite strong: for example, groups do not satisfy it. However, in the case of commutative associative algebras, the cosmash product is nothing more than the tensor product; which explains why in this case it is associative. We prove that in the setting of operadic varieties of algebras over a field, it is the only example. Further examples in the non-operadic case are also discussed.

math.RA↗

arXiv2020

Algebras with representable representations

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to \mathit{Der}(X)$ from $B$ to the Lie algebra $\mathit{Der}(X)$ of derivations on $X$. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field $\mathbb{K}$, in such a way that these generalised derivations characterise the $\mathbb{K}$-algebra actions. We prove that the answer is no, as soon as the field $\mathbb{K}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions -- which are usually called representations or Beck modules -- suffices for this to be true. Thus we characterise the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasises the unique role played by the Lie algebra of linear endomorphisms $\mathfrak{gl}(V)$ as a representing object for the representations on a vector space $V$.

math.RA↗