Reflections and Drinfeld twists for set-theoretic Yang–Baxter maps

The Yang–Baxter equation (ybe) and the reflection equation (re) both come from mathematical physics, and they can be defined in any monoidal category. For cartesian monoidal categories, we prove that every solution to the re provides a Drinfeld twist for a solution of the ybe. As we observe, Drinfeld twists of solutions are relevant for the following reason: two solutions to the ybe (in any strict monoidal category) are related by a Drinfeld twist, if and only if they induce equivalent representations of the braid group. In the category of sets, it is known that every solution is associated with a structure group, which is a braided group in the sense of Lu et al. (Duke Math J 104(1):1–18, 2000). Using De Commer’s notion of a braided action, we then define group reflections for a braided group. We prove that group reflections provide group Drinfeld twists in the sense of Ghobadi. Finally, we characterise when a reflection on a solution (X, r) can be extended to a group reflection on its structure group G(X, r).