A decomposition problem for involutive solutions to the Yang-Baxter equation

Abstract

An open question of Ramírez and Vendramin (IMRN, 2022) concerning the decomposability of a finite involutive non-degenerate solution to the set-theoretic Yang-Baxter equation is answered in the negative. Counterexamples are obtained from a class of singular, non-simple solutions which are not of finite primitive level. The ideals of the corresponding braces are calculated. For some of these braces it is proved that any minimal ideal gives rise to a simple solution to the Yang-Baxter equation. The concept of ``primitive level'' is extended to infinite solutions and shown to be a special case of a general concept of length. A quantitative interpretation of finite primitive level is obtained from a numeric invariant related to the augmentation ideal.