arXiv subjects

Ilaria Colazzo

Publications and source records attributed to Ilaria Colazzo.

13 recordsLinked to original sources

arXiv2026

On set-theoretic solutions of pentagon equation and positive basis Hopf algebras

We investigate the connection between bijective, not necessarily finite, set-theoretic solutions of the pentagon equation and Hopf algebras. Firstly, we prove that finite solutions correspond to Hopf algebras with the positive basis property. As a corollary we generalise Lu-Yan-Zhu classification to arbitrary characteristic $0$ fields $k$. Secondly, we study the general problem of when a Hopf algebra has a basis yielding a set-theoretic solution. Finally, we classify all (co)commutative bijective solutions. This result requires to obtain a description of all bases of a group algebra $k[G]$ yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings $A \rtimes N$ of $G$ with $A$ a finite abelian group.

math.RA↗

arXiv2024

On the cabling of non-involutive set-theoretic solutions of the Yang--Baxter equation

We extend the cabling method by Lebed, Ram\'irez and Vendramin from involutive to bijective non-degenerate set-theoretic solutions of the Yang--Baxter equation by working in the Yang--Baxter monoid $M(X,r)$ rather than the group $G(X,r)$. This shift in approach overcomes the obstruction that, for non-involutive solutions, the canonical map from $X$ to the Yang--Baxter group $G(X,r)$ need not be injective and yields a well-defined cabling. We prove that cabling is functorial on biquandles and that the diagonal map transforms as $q\mapsto q^k$. We also show that decomposability is preserved by injectivization and by passing to the associated biquandle, allowing us to work within that class without loss of generality. This leads to criteria for (in)decomposability. As an application, we obtain that square-free solutions with nilpotent derived monoid are decomposable.

math.QA↗

arXiv2024

Skew bracoids containing a skew brace

Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A consequence of this correspondence is that skew bracoids in this family can be used to obtain and study solutions of the set-theoretic Yang--Baxter equation; we study this process and the resulting solutions. We give numerous examples of skew bracoids satisfying our hypothesis, drawing upon a variety of constructions in the literature.

math.RA↗

arXiv2023

Simple solutions of the Yang-Baxter equation

We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach includes and extends the work of Joyce for quandles and Castelli for involutive solutions, demonstrating that the simplicity of a solution can be understood through its associated permutation skew left brace. In particular, we show that this skew left brace must have the smallest non-zero ideal, and the quotient by this ideal gives a trivial skew left brace of cyclic type; clearly all simple skew left braces satisfy these assumptions. As an application of our approach we construct and characterise new infinite families of simple solutions that are neither involutive nor quandles. Additionally, we show that our method can be applied to simple skew left braces to generate further families of simple solutions.

math.RA↗

arXiv2023

On derived-indecomposable solutions of the Yang--Baxter equation

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.

math.GR↗

arXiv2023

Structure algebras of finite set-theoretic solutions of the Yang--Baxter equation

Quadratic algebras related to some classes of finite left non-degenerate solutions (X,r) of the Yang--Baxter equation have been intensively studied since they are the associative ring-theoretical tool to study solutions. These are the monoid algebras K[M(X,r)] and K[A(X,r)], over a field K, of its structure monoid M(X,r) and left derived structure monoid A(X,r). In case r is bijective (and thus also right non-degenerate) it is known that these algebras are representable (hence PI), left and right Noetherian and have finite Gelfand-Kirillov dimension. Moreover, such algebras are domains (or equivalently prime) if and only if they have finite global dimension, which also is equivalent to r being an involutive map. In this paper we deal with structure algebras of arbitrary finite left non-degenerate solutions (X,r), except for the last section. If (X,r) satisfies additional conditions, such as being bijective, idempotent or left derived, it has been shown in a series of papers that K[M(X,r)] is left Noetherian. In the first part of the paper we show that the algebra K[M(X,r)] always is left Noetherian. Via divisibility by generators, we construct an ideal chain in M(X,r) that has very strong algebraic structural properties on its Rees factors. This allows to obtain characterizations of when the algebras K[M(X,r)] and K[A(X,r)] are right Noetherian. Intricate relationships between ring-theoretical and homological properties of these algebras and properties of the solution (X,r) are proven. Furthermore, we describe the cancellative congruences of A(X,r) and M(X,r) as well as the prime spectrum of K[A(X,r)]. This then leads to an explicit formula for the Gelfand-Kirillov dimension of K[M(X,r)] and it equals the classical Krull dimension.

math.RA↗

arXiv2022

Finite idempotent set-theoretic solutions of the Yang--Baxter equation

It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $φ_x$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps $φ_x$ are equal to an automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The structure algebra always is a left Noetherian representable algebra of Gelfand--Kirillov dimension one. To prove these results it is shown that the structure semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_u$ indexed by the diagonal, each $S_u$ has a group of quotients $G_u$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.

math.RA↗

arXiv2022

Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses

To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzeziński. Fundamental examples of YB-semitrusses are structure monoids of left non-degenerate set-theoretic solutions and (skew) left braces. Gateva-Ivanova and Van den Bergh introduced structure monoids and showed their importance (as well as that of the structure algebra) for studying involutive non-degenerate solutions. Skew left braces were introduced by Guarnieri, Vendramin and Rump to deal with bijective non-degenerate solutions. Hence, YB-semitrusses also yield a unified treatment of these different algebraic structures. The algebraic structure of YB-semitrusses is investigated, and as a consequence, it is proven, for example, that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective. Furthermore, it is shown that some finite left non-degenerate solutions can be reduced to non-degenerate solutions of smaller size. The structure algebra of a finitely generated YB-semitruss is an algebra defined by homogeneous quadratic relations. We prove that it often is a left Noetherian algebra of finite Gelfand-Kirillov dimension that satisfies a polynomial identity, but in general, it is not right Noetherian.

math.RA↗

arXiv2020

Set-theoretic solutions of the Pentagon Equation

A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ such that $s_{23}s_{13}s_{12}=s_{12}s_{23}$, where $s_{12}=s\times\mathrm{id}$, $s_{23}=\mathrm{id}\times s$ and $s_{13}=(τ\times\mathrm{id})(\mathrm{id}\times s)(τ\times\mathrm{id})$ are mappings from $S^3$ to itself and $τ\colon S^2\to S^2$ is the flip map, i.e., $τ(x,y) =(y,x)$. We give a description of all involutive solutions, i.e., $s^2=\mathrm{id}$. It is shown that such solutions are determined by a factorization of $S$ as direct product $X\times A \times G$ and a map $σ\colon A\to\mathrm{Sym}(X)$, where $X$ is a non-empty set and $A,G$ are elementary abelian $2$-groups. Isomorphic solutions are determined by the cardinalities of $A$, $G$ and $X$, i.e., the map $σ$ is irrelevant. In particular, if $S$ is finite of cardinality $2^n(2m+1)$ for some $n,m\geq 0$ then, on $S$, there are precisely $\binom{n+2}{2}$ non-isomorphic solutions of the Pentagon Equation.

math.RA↗

arXiv2020

Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang-Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

math.QA↗

arXiv2020

The algebraic structure of left semi-trusses

The distributive laws of ring theory are fundamental equalities in algebra. However, recently in the study of the Yang-Baxter equation, many algebraic structures with alternative "distributive" laws were defined. In an effort to study these "left distributive" laws and the interaction they entail on the algebraic structures, Brzeziński introduced skew left trusses and left semi-trusses. In particular the class of left semi-trusses is very wide, since it contains all rings, associative algebras and distributive lattices. In this paper, we investigate the subclass of left semi-trusses that behave like the algebraic structures that came up in the study of the Yang-Baxter equation. We study the interaction of the operations and what this interaction entails on their respective semigroups. In particular, we prove that in the finite case the additive structure is a completely regular semigroup. Secondly, we apply our results on a particular instance of a left semi-truss called an almost left semi-brace, introduced by Miccoli to study its algebraic structure. In particular, we show that one can associate a left semi-brace to any almost left semi-brace. Furthermore, we show that the set-theoretic solutions of the Yang-Baxter equation originating from almost left semi-braces arise from this correspondence.

math.QA↗

arXiv2019

The matched product of set-theoretical solutions associated with shelves

We investigate the matched product of solutions associated with right and left shelves. First, we prove that the requirements to provide the matched product of solutions that come from shelves can be simplified. Then we give conditions for left non-degeneracy of the matched product. Later, we compute the structure shelf of the matched product of solutions. Finally, we prove that the structure shelf of the matched product does not depend on the choice of the actions.

math.QA↗

arXiv2019

The matched product of the solutions to the Yang-Baxter equation of finite order

In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions $r_S$ and $r_T$ is of finite order if and only if $r_S$ and $r_T$ are. Furthermore, we show that with sufficient information on $r_S$ and $r_T$ we can precisely establish the order of the matched product. Finally, we prove that if $B$ is a finite semi-brace, then the associated solution $r$ satisfies $r^n=r$, for an integer $n$ closely linked with $B$.

math.QA↗