In this paper, the relations between the Yang–Baxter equation and affine actions are explored in detail. In particular, we classify the injective set-theoretic solutions of the Yang–Baxter equation in two ways: (i) by their associated affine actions of their structure groups on their derived structure groups, and (ii) by the $C^*$-dynamical systems obtained from their associated affine actions. On the way to our main results, several other useful results are also obtained.
"Affine actions and the Yang–Baxter equation." Adv. Oper. Theory 3 (3) 710 - 730, Summer 2018. https://doi.org/10.15352/aot.1801-1298