Silvia Properzi (Vrije Universiteit Brussel) – Skew Braces, the YBE, and some results in the classification of solutions

This talk explores the interplay between algebraic structures known as skew braces and set-theoretic solutions to the Yang–Baxter equation (YBE), a central object in mathematical physics and quantum group theory. Skew braces provide a powerful framework for constructing and analysing solutions.

We focus on involutive solutions, in particular on their permutation skew braces that allow for explicit computations and structural analysis.

We introduce the necessary background on skew braces and the YBE, we study the role of the permutation skew brace, which captures essential symmetries of solutions and serves as an effective invariant. As a concrete application, we present a full classification, up to isomorphism, of all involutive solutions of size $p^2$, where $p$ is a prime.