Skew braces and the YBE
Édouard Feingesicht (Caen)
Germs and Sylows for structure group of solutions to the Yang-Baxter equation
Abstract
In 1992, Drinfeld addressed the question of classifying set-theoretical solutions to the Yang-Baxter equation. Since, many advances have been made, in particular through the introduction and study of a group associated to a solution : the structure group. In this talk, we will present the use of a monomial representation to study those groups, allowing for intuitive and simple proofs of some well-known results (I-structure, Garsideness, …), then we will focus on the associated Dehornoy’s class and germs (playing a role similar to the one of Coxeter groups for spherical Artin groups). In particular, we will discuss the values of Dehornoy’s class, and state a result on the Sylows of the germs, which reduces the classification problem to particular solutions (those with class a power of a prime)
Raúl Sastriques (Valencia)
Decomposability theorems for the YBE
Abstract
In relation to the Garside groups, we find the Yang-Baxter equation, which provides a family of Garside groups of great interest. The purpose of the talk will be to explain the relation between the two structures established by F. Chouraqui, as well as our latest results, regarding certain arithmetic properties, which provide new criteria for the decomposability of these type of solutions.
Andrew Darlington (Exeter)
Things to see and do with subgroups of the holomorph
Abstract
This talk will begin by introducing skew braces and Hopf-Galois structures on Galois extensions, followed by exploring the connection between them. On the Hopf-Galois side, it’s possible to not only look at regular subgroups of the holomorph, but also the more general notion of transitive subgroups, which now correspond with separable extensions (where they don’t necessarily have to be normal). This generalisation leads to a number of very interesting questions that we can ask, some of which will be covered in the second half of the talk.
Ilaria Colazzo
Combinatorial solutions to the Yang-Baxter equation and applications
Abstract
The first part of the talk will connect the topological problem of distinguishing mathematical knots with combinatorial solutions to the Yang-Baxter equation. The second one will be devoted to a new algebraic structure (skew braces) that provides the perfect tool to study and classify such solutions.