Free Skew Braces and Free Solutions of the Yang--Baxter Equation

Abstract

We prove that the structure skew brace of a free non-degenerate bijective solution to the Yang--Baxter Equation is free (a similar result is obtained in a category of nilpotent solutions, denoted by $\mathcal{RNS}_n$). We offer a workable construction of the free right nilpotent skew braces of arbitrary class which allows us to prove (among many other things) that this free object has free additive/multiplicative groups, and that under certain additional circumstances it must also be residually finite and Hopfian. The main consequences of this construction are that the free non-degenerate bijective solution to the Yang--Baxter Equation in the catego\-ry~$\mathcal{RNS}_n$ has a solvable Word Problem, and that every law holding in for finite solution of the previous type also holds for {\it every} solutions of the same type. In the remainder of the paper, we present further explicit realizations of free objects and explore their consequences. Among these are free two-sided skew braces of abelian type (with abelian multiplicative group) and free centrally nilpotent skew braces of class $2$.