Indecomposability of solutions to the Yang--Baxter equation via skew braces

The Yang–Baxter equation (YBE), originating in statistical mechanics and central in quantum group theory, admits a combinatorial version in terms of set-theoretic solutions introduced by Drinfeld. Their classification is a difficult problem, with complete results known only in special cases. This motivates the study of indecomposable solutions, which can be regarded as fundamental building blocks. Skew braces provide a natural algebraic framework for constructing and analysing set-theoretic solutions. In this talk, we focus on indecomposable involutive solutions and explain how structure groups and permutation skew braces encode their properties and allow us to detect indecomposability. As an illustration, we discuss classification results in small size and indicate how Dehornoy’s monomial representations reflect these structural features.

May 28, 2026

Algebra seminar