arXiv subjects

Manuel Mancini

Publications and source records attributed to Manuel Mancini.

13 recordsLinked to original sources

arXiv2026

Weak action representability of 2-nilpotent groups

In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by determining a universal strict general actor of an object $X$, which turns out to be the group $\operatorname{Aut}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B \to \operatorname{Aut}_c(X)$ that define an action of $B$ on $X$ in $\mathsf{Nil}_2(\mathsf{Grp})$. We then show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not action representable, and that the existence of a weak representation is related to the amalgamation property. Using the construction of an amalgam of a suitable family of abelian subgroups of $\operatorname{Aut}_c(X)$, we prove that the category $\mathsf{Nil}_2(\mathsf{Grp})$ is weakly action representable, and that a weak representing object can be chosen to be an abelian group. Finally, we show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not locally algebraically cartesian closed.

math.CT↗

arXiv2026

On actions and split extensions in varieties of hoops: the case of strong section

The aim of this article is to investigate internal actions and split extensions in the variety of hoops. We provide a characterization of split extensions with strong section in terms of strong external actions. Beyond the general setting of hoops, the study is extended to the subvarieties of basic hoops, Wajsberg hoops, Gödel hoops and product hoops. Within the setting of basic hoops and their bounded counterparts, BL-algebras, the double negation yields a significant example of split extension with strong section, thus motivating our approach. A connection between strong external actions of hoops and the semidirect product construction introduced by W. Rump in the cateogory of L-algebras is established.

math.CT↗

arXiv2025

On Lie-holomorphs of Leibniz algebras

We study the notion of the Lie-holomorph of a Leibniz algebra, recently introduced by N. P. Souris as a generalisation of the classical holomorph construction for Lie algebras. We establish a connection between the Lie-holomorph construction and the Leibniz algebra of biderivations defined by J.-L. Loday, and we prove that a linear endomorphism is a Lie-derivation if and only if it is simultaneously a derivation and an anti-derivation. As an application, we classify the Lie-holomorph algebras of all low-dimensional non-Lie Leibniz algebras over a field of characteristic different from $2$.

math.RA↗

arXiv2025

Coherent and ideal actions in ideally exact categories

In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.

math.CT↗

arXiv2025

On the representability of actions of Leibniz algebras and Poisson algebras

In a recent paper, motivated by the study of central extensions of associative algebras, G. Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras by describing explicitly a universal strict general actor.

math.CT↗

arXiv2025

Action accessible and weakly action representable varieties of algebras

The main goal of this article is to investigate the relationship between action accessibility and weak action representability in the context of varieties of non-associative algebras over a field. Specifically, using an argument of J. R. A. Gray in the setting of groups, we prove that the varieties of $k$-nilpotent Lie algebras ($k \geq 3$) and the varieties of $n$-solvable Lie algebras ($n \geq 2$) do not form weakly action representable categories. These are the first known examples of action accessible varieties of non-associative algebras that fail to be weakly action representable, establishing that a subvariety of a (weakly) action representable variety of non-associative algebras needs not be weakly action representable. Eventually, we refine J. R. A. Gray's result by proving that the varieties of $k$-nilpotent groups ($k \geq 3$) and that of $2$-solvable groups are not weakly action representable.

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↗

arXiv2024

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra $\mathfrak{g}$ with $\operatorname{dim}[\mathfrak{g},\mathfrak{g}]=1$ is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra $\mathfrak{l}_{2n+1}^{J_1}$, where $J_1$ is the $n \times n$ Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

math.RA↗

arXiv2023

Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by $L_n$, where $n=\dim_{\mathbb{F}} L_n$. This generalizes the result found in [11], which is only valid when $\mathbb{F}=\mathbb{C}$. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$ by integrating it into a Lie rack.

math.RA↗

arXiv2023

Biderivations of low-dimensional Leibniz algebras

In this paper we give a complete classification of the Leibniz algebras of biderivations of right Leibniz algebras of dimension up to three over a field $\mathbb{F}$, with $\operatorname{char}(\mathbb{F})\neq 2$. We describe the main properties of such class of Leibniz algebras and we also compute the biderivations of the four-dimensional Dieudonné Leibniz algebra $\mathfrak{d}_1$. Eventually we give an algorithm for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis.

math.RA↗

arXiv2023

Derivations of two-step nilpotent algebras

In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation.

math.RA↗

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↗

arXiv2021

Two-step nilpotent Leibniz algebras

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. In particular, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical $(2n+1)-$dimensional Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. We also show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group.

math.RA↗