The Algebra Behind the Yang–Baxter Equation

The Yang–Baxter equation (YBE), originally introduced in statistical mechanics and later playing a key role in quantum group theory, is a fundamental equation describing a consistency condition for interacting systems. In the 1990s, Drinfeld proposed the study of set-theoretic solutions of the YBE, giving rise to a more combinatorial version of the problem that has since developed into an active area of research. Understanding and classifying these solutions remains a subtle task. A key insight of the past decade is that skew braces provide a natural algebraic framework that governs set-theoretic solutions of the YBE. These structures encode a compatibility between two group operations on the same set, and allow solutions to be constructed and analysed in a systematic way. This perspective provides a bridge between the combinatorics of solutions and classical group-theoretic methods, allowing structural questions to be approached in an algebraic way. In this talk, I will introduce skew braces and explain how they give rise to solutions of the YBE. I will focus in particular on involutive solutions, where the correspondence is especially transparent, and explain how structure and permutation skew braces capture their algebraic structure and lead to classification results. As an illustration, I will briefly discuss classification results in small size. If time permits, I will also indicate how further algebraic constructions, including representation-theoretic ones, reflect key properties of the underlying solutions.

May 12, 2026