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 ).