Degenerate involutive set-theoretic solutions to the Yang-Baxter equation

Abstract

Degenerate solutions to the Yang-Baxter equation are studied by means of associated semibraces and groups. Using a characterization in terms of cycle sets, a non-degenerate part is separated from a purely degenerate one. More precisely, any cycle set X has a distinguished non-degenerate sub-cycle set X×. If X× is trivial, the permutation group G(X) of X admits a natural partial order, with a strong semibrace S(X) as its negative cone, so that G(X) is a group of left fractions of S(X). The semibrace S(X) is constructed exactly like the self-similar closure of an L-algebra. It is proved that each non-trivial Garside group (e.g., Artin's braid group) gives rise to a degenerate cycle set. With a graded algebra related to the first Weyl algebra, a negative answer to a recent problem of Bonatto et al. (2021) [2] is obtained.