arXiv subjects

Gianmarco La Rosa

Publications and source records attributed to Gianmarco La Rosa.

11 recordsLinked to original sources

arXiv2026

Molnár's Theorem for split $C^*$-by-nilpotent algebras

Kubo-Ando means are a rich class of important binary operations on the set of positive elements of an operator algebra. The aim of this article is to understand the algebraic structure of these operations. We generalize a recent result of L. Molnár for $C^*$-algebras to the class of split $C^*$-by-nilpotent algebras by using the method of analytic local expansion of binary operations. Eventually, we prove that in any $C^*$-algebra $\mathcal{A}$, the identity $[x,[y,z]]=0$ implies commutativity.

math.OA↗

arXiv2026

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↗

arXiv2026

Fuzzy OWL 2 Reasoning: A Re-Engineered Python Framework

In many real-world domains, knowledge is inherently vague or imprecise - features that classical ontology languages, based on crisp Description Logics (DLs), are unable to capture. This shortcoming poses particular challenges for applications in the Semantic Web and Explainable Artificial Intelligence (XAI), where robust reasoning over graded information is essential. Fuzzy ontologies address this limitation by enriching DLs with fuzzy logic, enabling the expression of partial truth and supporting more nuanced modelling of real-world knowledge. We present fuzzy-dl-owl2, a complete re-engineering in Python of the fuzzyDL reasoner and the Fuzzy OWL 2 framework. The former is an expressive fuzzy DL reasoner, while the latter allows for defining fuzzy ontologies within OWL 2. Our contribution addresses several shortcomings of the original software, including semantic inconsistencies, rigid architectural design, and limited solver integration. The re-implementation features a modular class hierarchy tailored for extensibility, supports a broader range of Mixed-Integer Linear Programming (MILP) solvers (including open-source alternatives), and corrects IRI ambiguities arising from overlapping ontological elements. Furthermore, a dedicated Python library (pyowl2) has also been developed to handle OWL 2 annotations in a standards-compliant manner, improving interoperability with existing Semantic Web tooling and resolving IRI ambiguities. The resulting framework offers a portable, extensible, and theoretically grounded platform for reasoning with fuzzy ontologies, suitable for both research and deployment in vague-aware systems. Performance tests have also been conducted that show improved execution times w.r.t. the original Java implementation. The source code and full documentation are publicly available to facilitate community adoption and further development.

math.GM↗

arXiv2025

Biderivations of complete Leibniz algebras

If one wishes to define a complete Leibniz algebra in such a way as to extend the notion of a complete Lie algebra, two distinct definitions can be found in the current literature. Since biderivations on complete Lie algebras have already been studied, in order to extend those results and considering that Leibniz algebras are, among others, a natural generalisation of Lie algebras, we study here the biderivations of complete Leibniz algebras according to both definitions. In each case, we provide necessary and sufficient conditions for a bilinear map to be a biderivation of a Leibniz algebra. Finally, we analyse both symmetric and skew-symmetric biderivations, highlighting their structural properties.

math.RA↗

arXiv2025

A Lie Bracket on the Space of (Right) Biderivations of a Lie Algebra

Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms. Motivated by this, we study biderivations-bilinear maps capturing higher-order infinitesimal symmetries. This work focuses on right biderivations of Lie algebras, introducing Lie brackets on spaces of biderivations to explore their algebraic and geometric properties. We analyse the interplay between left and right biderivation brackets through symmetric and skew-symmetric cases, providing a coherent Lie algebra framework. Moreover, we initiate the construction of Lie groups corresponding to the Lie algebra of biderivations, linking infinitesimal and global structures. Our results offer new perspectives on higher-order derivations with potential applications in deformation theory and generalised symmetry studies.

math.RA↗

arXiv2025

A New Definition of Superbiderivations for Lie Superalgebras

In this paper, we study superbiderivations on Lie superalgebras from structural and geometric perspectives. Motivated by the classical fact that the bracket of a Lie algebra is itself a biderivation, we propose a new definition of superbiderivation for Lie superalgebras, one that requires the bracket to be a superbiderivation, a condition not satisfied by existing definitions in the literature. Our focus is on complete Lie superalgebras, a natural generalization of semisimple Lie algebras that has emerged as a promising framework in the search for alternative structural notions. In this setting, we introduce and study linear supercommuting maps, comparing our definition with previous proposals. Finally, we present two applications: one involving the superalgebra of superderivations of the Heisenberg Lie superalgebra and another offering initial geometric insights into deformation theory via superbiderivations.

math.RA↗

arXiv2024

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↗

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

Biderivations of complete Lie algebras

The authors of this article intend to present some results obtained in the study of biderivations of complete Lie algebras. Firstly they present a matricial approach to do this, which was a useful and explanatory tool not only in the study of biderivations but also in the synthesis of these results. Then they study all biderivations of a Lie algebra $L$ with $\operatorname{Z}(L)=0$ and $\operatorname{Der(L)}=\operatorname{ad(L)}$, called complete. Moreover, as an application of the previous result, they describe all biderivations of a semisimple Lie algebra (that are complete), extending a result obtained by X. Tang in ([20]) that describes all biderivations of a complex simple Lie algebra. And thirdly, results on symmetric and skew-symmetric biderivations are also presented.

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

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↗