Proof of Rump's Retraction Conjecture for Quasilinear Cycle Sets

Nondegenerate cycle sets were introduced by Rump as an algebraic framework for nondegenerate, involutive solutions to the Yang–Baxter equation. Nondegenerate cycle set structures on abelian groups, such as translation-invariant and quasilinear cycle sets, are of particular interest when studying the retraction problem in the theory of the Yang–Baxter equation. In this article, we solve the retraction problem for finite quasilinear cycle sets by showing that each nontrivial quasilinear cycle set is retractable, thus proving a conjecture of Rump.