arXiv subjects

Davide Ferri

Publications and source records attributed to Davide Ferri.

7 recordsLinked to original sources

arXiv2026

On quiver skew braces, their ideals and products

Quiver skew braces or skew bracoids are equivalent to braided groupoids, that is, groupoids with a constraint of abelianity. They are the quiver-theoretic version of skew braces, an increasingly studied structure lying in the intersection of group and ring theory. In this paper, we define ideals and quotients for quiver skew braces, with respect to two notions of morphisms. Following the track of a previous work of ours (2025), we define a classical semidirect product \`a la Brown, and a categorical semidirect product \`a la Bourn and Janelidze, for the category of quiver skew braces. It is known that connected groupoids can be expressed as the datum of a group of loops and a set of vertices. We demonstrate how no such decomposition holds for quiver skew braces, which makes their theory richer than the theory of groupoids.

math.RT↗

arXiv2025

Split Lemma and First Isomorphism Theorem for groupoids

Groupoids are the oidification of groups, and they are largely used in topology and representation theory. We consider here the category $\mathsf{Gpd}$ of all groupoids with all morphisms, and the category $\mathsf{Gpd}_\Lambda$ of groupoids over a fixed set of vertices $\Lambda$, with morphisms fixing $\Lambda$. In $\mathsf{Gpd}_\Lambda$, a First Isomorphism Theorem is already well known; see \'Avila, Mar\'in, and Pinedo (2020). Famously, the First Isomorphism Theorem fails to hold in $\mathsf{Gpd}$. However, we retrieve here a universally lifted version of the First Isomorphism Theorem in $\mathsf{Gpd}$, through the definition of virtual kernels. Semidirect products of a group by a groupoid are well known. We define crossed products in $\mathsf{Gpd}$, and prove that they are equivalent to split epimorphisms, i.e. that they are the `categorial' notion of semidirect product in $\mathsf{Gpd}$ in the sense of Bourn and Janelidze (1998). We observe that in $\mathsf{Gpd}_\Lambda$ crossed products and semidirect products are essentially equivalent, under mild assumptions, and our Split Lemma in $\mathsf{Gpd}$ collapses to a much simpler Split Lemma in $\mathsf{Gpd}_\Lambda$ that appears in Metere and Montoli (2010) and Ibort and Marmo (2023).

math.GR↗

arXiv2025

Reflections and Drinfeld twists for set-theoretic Yang-Baxter maps

The Yang-Baxter equation (YBE) and the reflection equation (RE) both come from mathematical physics, and they can be defined in any monoidal category. For cartesian monoidal categories, we prove that every solution to the RE provides a Drinfeld twist for a solution of the YBE. As we observe, Drinfeld twists of solutions are relevant for the following reason: two solutions to the YBE (in any strict monoidal category) are related by a Drinfeld twist, if and only if they induce equivalent representations of the braid group. In the category of sets, it is known that every solution is associated with a structure group, which is a braided group in the sense of Lu, Yan, and Zhu (2000). Using De Commer's notion of a braided action, we then define group reflections for a braided group. We prove that group reflections provide group Drinfeld twists in the sense of Ghobadi. Finally, we characterise when a reflection on a solution (X,r) can be extended to a group reflection on its structure group G(X,r).

math.QA↗

arXiv2025

Structure groupoids of quiver-theoretic Yang-Baxter maps

Solutions to the quiver-theoretic quantum Yang-Baxter equation are associated with structure categories and structure groupoids. We prove that the structure groupoids of involutive non-degenerate solutions are Garside. This generalises a well-known result about the structure groups of set-theoretic solutions, due to Chouraqui. We also construct involutive non-degenerate solutions from suitable presented categories. We then investigate the case of solutions of principal homogeneous type. Finally, we present some examples of this new class of Garside groupoids.

math.QA↗

arXiv2024

On dynamical skew braces and skew bracoids

Dynamical skew braces are known to produce solutions to the quiver-theoretic Yang--Baxter equation. Under a technical hypothesis, we prove that these solutions are braided groupoids (and hence skew bracoids in the sense of Sheng, Tang and Zhu). Conversely, every connected braided groupoid can be parallelised, making it isomorphic to a dynamical skew brace. We study the combinatorics of these objects, depending on some strings of integer invariants.

math.QA↗

arXiv2024

Matched pairs and Yetter-Drinfeld braces

It is proven that a matched pair of actions on a Hopf algebra $H$ is equivalent to the datum of a Yetter-Drinfeld brace, which is a novel structure generalising Hopf braces. This improves a theorem by Angiono, Galindo and Vendramin, originally stated for cocommutative Hopf braces. These Yetter-Drinfeld braces produce Hopf algebras in the category of Yetter-Drinfeld modules over $H$, through an operation that generalises Majid's transmutation. A characterisation of Yetter-Drinfeld braces via 1-cocycles, in analogy to the one for Hopf braces, is given. Every coquasitriangular Hopf algebra $H$ will be seen to yield a Yetter-Drinfeld brace, where the additional structure on $H$ is given by the transmutation. We compute explicit examples of Yetter-Drinfeld braces on the Sweedler's Hopf algebra, on the algebras $E(n)$, on $\mathrm{SL}_{q}(2)$, and an example in the class of Suzuki algebras.

math.QA↗

arXiv2019

Energy conversion processes with perovskite-type materials

Mixed oxides derived from the perovskite structure by combination of A- and B-site elements and by partial substitution of oxygen provide an immense playground of physico-chemical properties. Here, we account for own research conducted at the Paul Scherrer Institute on perovskite-type oxides and oxynitrides used in electrochemical, photo(electro)chemical and catalytic processes aiming at facing energy relevant issues.

cond-mat.mtrl-sci↗