Skew braces, indecomposability and Dehornoy’s representations for solutions to the Yang-Baxter equation

This talk explores the interplay between algebraic structures known as skew braces and set-theoretic solutions to the Yang–Baxter equation (YBE). Skew braces provide a powerful framework for constructing and governing such solutions. We focus on involutive cases, examining their structure groups, Coxeter-like groups, and permutation skew braces, which enable explicit computations and structural analysis. After reviewing the necessary background on skew braces and the YBE, we discuss how the permutation skew brace captures essential symmetries of a solution and serves as an effective invariant. As an illustration, we briefly outline the classification of involutive solutions of size $p^2$, where $p$ is a prime. Finally, using the brace structure of the associated Coxeter-like groups, we show how the properties of these groups - and, in particular, Dehornoy’s monomial representations - reflect the indecomposability of the underlying solutions.

Oct 28, 2025