The *“ Algebra Reading Seminar*” group usually meets weekly to discuss current papers related to algebraic structures linked to solutions of the Yang-Baxter equation.

Open access is allowed to everyone is interested in attending seminars. Requirements will be specified for each seminar.

**Organizers**: prof. Francesco Catino and Ilaria Colazzo.

Il gruppo “* Algebra Reading Seminar*” si incontra per discutere gli ultimi articoli relativi a strutture algebriche correlate con le soluzioni dell’equazione di Yang-Baxter.

Chiunque sia interessato agli argomenti trattati può partecipare. I prerequisiti saranno specificati per i singoli seminari.

A cura del prof. Francesco Catino e Ilaria Colazzo.

**A new construction of solutions to the Yang-Baxter equation**

May 29, 2019

11.00 – 13.00 – Aula Seminari*Paola Stefanelli*

The pentagon equation (PE) is a basic equation of Mathematical Physics.

Recent developments in this field of investigations can be found in [1]. In this talk, we show a new application of the PE. In particular, the PE turns out to be a useful tool to find set-theoretical solutions of the well-known Yang-Baxter equation.

[1] F. Catino, M. Mazzotta, M.M. Miccoli: Set-theoretical solutions of the åpentagon equation on groups, Preprint available at arXiv:1902.04310.

*Shelves and the matched product of solutions of the Yang-Baxter equation*

May 15, 2019

11.00 – 13.00 – Aula Seminari*Ilaria Colazzo*

In this talk, we focus on the matched product of solutions associated with shelves, and we simplify the conditions that allow obtaining this product. Furthermore, we study the properties of the shelf related to the matched product of two left non-degenerate solutions.

*About a question of Gateva-Ivanova and Cameron*

April 10, 2019

11.00 – 13.00 – Aula M5*Marco Castelli*

In 2011 Gateva-Ivanova and Cameron [2], following a suggestion due by P. Martin, posed the following question:

**Question.** For each positive integer denote by the minimal integer so that there exists a square-free involutive multipermutational solution of order and with . How does depend on ?

They showed [2] that for every and, some years later, Lebed and Vendramin improve [3] the estimation showing indirectly that for every . In this talk we will present some recent results obtained in [1] that allow us to improve the estimation of the sequence .

*Nuove costruzioni di soluzioni insiemistiche dell’equazione di Yang-Baxter*

April 4, 2019

11.00 – 13.00 – M5*Francesco Catino*

Partendo da alcuni esempi, si indicheranno alcune piste di lavoro per la costruzione di nuove soluzioni insiemistiche dell’equazione di Yang-Baxter.

*On the matched product of set-theoretical solutions*

March 20, 2019

11.00 – 13.00 – M5*Paola Stefanelli*

The problem of finding set-theoretical solutions of the Yang-Baxter equation [2] (shortly solutions) has been dealt with from different points of view in the last years.

In [1] a novel construction technique for solutions was developed, the *matched product*. In this talk, we will display a method to construct new classes of particular solutions through the matched product of other solutions in the same class. As a particular case, starting from involutive (resp. idempotent) solutions the matched product is still involutive (resp. idempotent), and vice versa.

[1] F. Catino, I. Colazzo, P. Stefanelli, *The matched product of set-theoretical solutions of the Yang-Baxter equation*, submitted.

[2] V.G. Drinfeld, *On some unsolved problems in quantum group theory*. Lecture Notes in Math. 1510 (1992) 1-8.

*Indecomposable cycle sets with cyclic permutation group*

March 13, 2019

11.00 – 13.00 – M5*Giuseppina Pinto*

In order to study the indecomposable cycle sets, a possible approach involves the permutation group associated to the cycle sets.

In this talk we consider the particular case of indecomposable cycle sets with cyclic permutation group. We will introduce, in this particular case, the definition of the Socle of a cycle sets and we will see how an indecomposable cyle sets with cyclic permutation group can be pictured.

[1] M. Castelli, G. Pinto, Indecomposable involutive set-theoretic solutions with cyclic permutation group, *work in progress*.

*The Retraction Problem for quasi-linear left cycle sets*

March 6, 2019

11.00 – 13.00 – Aula Seminari*Marco Castelli*

In this talk we will discuss on a particular class of left cycle sets called *quasi-linear*. We will see briefly the basic theory and we will discuss on a problem posed by Rump in [2] and the related partial results obtained in [1].

[1] M. Castelli, F. Catino, M.M. Miccoli and G. Pinto, Dynamical
extensions of quasi-linear left cycle sets and the Yang-Baxter equation,
* Journal Algebra Appl.* (to appear).

[2] W. Rump,
Quasi-Linear Cycle Sets and the Retraction Problem for Set-Theoretic
Solutions of the Quantum Yang-Baxter Equation, *Algebra**Colloq.***23 **(1) (2016) 149-166.

*Application of the Yang-Baxter equation to particle collisions*

February 27, 2019

11.00 – 13.00 – F2*Giuseppe Rizzello*

The set-theoretical Yang-Baxter equation presents a form, called two-parameter set-theoretical Yang-Baxter equation, where the solutions depends precisely from two parameters. This formulation can be used in the study of systems of particles which collide among themselves, since from the conservation laws of this system we can derive the velocities after the collisions in terms of their masses (the parameters) and their velocities before the collisions and obtain this way a solution of said equation. This fact allow us to examine the integrability (that is the solvability) of a particular system periodic sequences of collisions.

*Structure monoid of left non-degenerate solutions to the Yang—Baxter equation*

February 20, 2019

11.00 – 13.00 – Aula Benvenuti*Arne Van Antwerper*

(joint work with Eric Jespers and Lukasz Kubat)

In this talk we will discuss the structure monoid of a finite left non-degenerate solution of the Yang–Baxter equation and its associated monoid algebra. Using a realization of Lebed and Vendramin of as a regular submonoid in the semidirect product , we will prove that is a finite module over a central affine subalgebra. In particular, it is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension. Furthermore, we will discuss some results on prime ideals of and we will discuss the positive answer to a conjecture of Gateva–Ivanova on cancellativity of .

*Relation between braces and units of group rings*

February 13, 2019

11.00 – 13.00 – Aula Benvenuti*Charlotte Verwimp*

Finding solutions of the Yang-Baxter equation has been the subject of many studies in pure mathematics and theoretical physics. In particular, as proposed by Drinfeld [1], the study of set-theoretic solutions has been very attractive over the past few years. One of the tools to study these solutions, called braces, were introduced by Rump [2]. In this talk, we want to show another useful application of braces. In particular we describe a relation between braces and units of group rings [3]. In joint work with Lukasz Kubat, we try to generalize this result for skew braces, a generalization of braces. To do so, we define group near-rings, a generalization of group rings.

[1] V.G. Drinfeld. On some unsolved problems in quantum group theory. *Lecture Notes in Math*, 1510:1-8, 1992.

[2] W. Rump. Braces, radical rings, and the quantum Yang-Baxter equation. *J. Algebra*, 307:153-170, 2007.

[3] Y.P. Sysak. Product of groups and the quantum Yang-Baxter equation. *Notes of a talk in Advances in Group Theory and Applications*, 2011, Porto Cesareo.

*Matched product of solutions and semi-braces*

February 08, 2019

11.00 – 13.00 – Aula Benvenuti*Ilaria Colazzo*

The study of the set-theoretical solutions of the Yang-Baxter equation (simply solutions) was introduced in [Drinfeld, 1992]. First, we will introduce a novel solutions construction technique called the **matched product of solutions**. Then we will give some interesting results which show that the matched product is a useful tool to construct solutions with specific proprieties, such as non-degeneracy and involutivness. Finally, we will introduce the definition of the matched product of two semi-braces and we will see the relationship between matched product of solutions and matched product of semi-braces.

**Soluzioni dell’equazione pentagonale**

31 gennaio 2019

ore 11.00-12.00 – Aula Benvenuti*Marzia Mazzotta*

In questo incontro introdurremo le soluzioni insiemistiche dell’equazione pentagonale. A partire da una problematica che emerge nell’articolo di Kashaev e Reshetikhin [R.M. Kashaev, N. Reshetikhin, Symmetrically Factorizable Groups and Set-theoretical Solutions of the Pentagon Equation, Contemp. Math. 433 (2007), 267 – 279], esamineremo, in particolare, il caso di quelle definite su un gruppo.