Solutions of the Yang-Baxter equation associated to skew left braces, with applications to racks

© 2018 World Scientific Publishing Company. Given a skew left brace B, a method is given to construct all the non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation such that the associated permutation group (X,r) is isomorphic, as a skew left brace, to B. This method depends entirely on the brace structure of B. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.