Almost simple braces and primitive solutions to the Yang-Baxter equation

Non-degenerate involutive set-theoretic solutions to the Yang Baxter equation are known to be equivalent to non-degenerate cycle sets. Each cycle set has a structure group acting on it. If this action is primitive, the cycle set is said to be primitive. It is shown that retractable primitive cycle sets are of torsion type and belong to a small list which has been found previously by Cedo et al. [9] in the finite case. For irretractable primitive torsion cycle sets X, it is proved that X generates a canonical brace A such that |pX | = 1 for a unique prime p, and that the p-primary part of A is cofinite. The brace A is almost simple in the sense that A has a unique minimal non-zero brace ideal A(2) such that A/A(2) is a cyclic brace. Furthermore, it shown that the elements a is an element of A with pa = 0 form an F-p-linear brace ideal A[p] such that the adjoint group of A[p] has a trivial centre, explaining the finite case from a more general point of view. (c) 2022 Elsevier Inc. All rights reserved.