Affine actions and the Yang-Baxter equation

Document Type: Original Article



‎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.