Algebra Days in Caen 2022: from Yang–Baxter to Garside

March 24–25 2022

This issue of the series Algebra Days in Caen is devoted to set-theoretic solutions to the Yang–Baxter equation, and more specifically to the structure groups of such solutions and finite Coxeter-like quotients thereof, and to various interactions with Garside structures.

The conference is planned to be in-person.


  • Adolfo Ballester-Bolinches (Valencia)
  • Fabienne Chouraqui (Haifa)
  • Edouard Feingesicht (Caen)
  • Přemysl Jedlička (Prague)
  • Leandro Vendramin (Brussels)

Practical information

To register, please follow this link before March 10.

If you have any questions, please contact the organisers, Eddy Godelle and Victoria Lebed.

We can provide financial support to junior participants. Please contact the organisers if you are interested.

Some hotels in Caen: Hôtel du Château, Hôtel des Quatrans, Hôtel la Fontaine.

Conference venue: Université de Caen Normandie, Campus 2, building Sciences 3, Lecture Hall S3 044 + LMNO. To get there, you may follow these instructions.

The list of participants is available here.

The conference dinner will take place in the restaurant L'Aromate at 19:00.

Some braids in Caen. Image credit:


Thursday, March 24

13:30 Welcome and registration
14:00 Leandro Vendramin
15:00 Edouard Feingesicht
16:00 Coffee break
16:30 Fabienne Chouraqui
19:00 Conference dinner

Friday, March 25

9:30 Přemysl Jedlička
10:30 Coffee + poster session
11:30 Adolfo Ballester-Bolinches

Titles and abstracts

Adolfo Ballester-Bolinches: On Yang–Baxter Groups

A group is said to be an involutive Yang–Baxter group, or simply an IYB-group, if it is isomorphic to the permutation group of an involutive, non-degenerate set-theoretic solution of the Yang–Baxter equation. In this paper, Yang–Baxter groups (YB-groups for short) associated with not necessarily involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation are introduced and studied. Sufficient conditions for a group that can be factorised as a product of two YB-groups to be a YB-group are provided. Some earlier results for finite IYB-groups are also generalised for arbitrary (non-necessarily finite) YB-groups.

Fabienne Chouraqui: The Yang–Baxter equation, braces and Thompson's group F

We define non-degenerate involutive partial solutions as an extension of non-degenerate involutive set-theoretical solutions of the quantum Yang–Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but rather a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Edouard Feingesicht: An introduction to Coxeter-like groups

This talk aim at explaining how structure groups of solutions of the Yang–Baxter equation admit a quotient that plays the same role as Coxeter groups do for Artin–Tits groups. To do so, we will begin by introducing the basic definitions of Garside monoids and of structure groups of solutions of the Yang–Baxter equation. We will then give an overview of approaches such as cycle sets or I-structures.

Přemysl Jedlička: Involutive solutions of the Yang–Baxter equation of multipermutation level 2 and their permutation groups

In this talk we focus on involutive solutions of the Yang–Baxter equation of multipermutation level 2. They fall into 2 classes: most of the solutions are self-distributive; they are decomposable into Lyubashenko's solutions, they have abelian permutation groups and they can be obtained by a combinatorial construction from a set of abelian groups and a matrix. A more interesting class is formed by those that are not self-distributive since the class contains all the decomposable solutions of multipermutation level 2. We give some examples of them and we show some properties of their permutation groups.

Leandro Vendramin: Left-ordered groups, Garside groups and structure groups of solutions

We will discuss several properties of structure groups of solutions to the YBE. For example: When are these groups Garside or left-ordered? When are these groups diffuse? When do they have the unique product property? What can be said about Kaplansky's problems on group algebras over structure groups of solutions? Regarding these and similar questions, we will present results (and maybe even short proofs), examples and open problems.


Marzia Mazzotta: Inverse semi-braces and the Yang–Baxter equation


Vicent Pérez-Calabuig: On left and right nilpotency for skew left braces: a Jodan–Hölder like theorem

Joint work with Adolfo Ballester-Bolinches and Ramon Esteban-Romero.

Paola Stefanelli: YBE solutions of pentagonal type


Arne Van Antwerpen: Monoids of left non-degenerate solutions of YBE


Financial support:

Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie, CNRS, GDR Tresses, GDR Topologie Algébrique et Applications, Fédération Normandie Mathématiques