Groupoid intertwiner and twist for dynamical Yang--Baxter equation: part I

Intertwiner is a homomorphism between two existing dynamical R matrices, first introduced by Baxter in eight vertex-SOS correspondence, we develop certain equivalence relations among R matrices using intertwiners. Twist is a homomorphism that twist a dynamical R matrix to get a new dynamical R matrix, we introduce a kind of notion of twist that generalize classical Drinfeld twist in quasi-triangular Hopf algebra and some dynamical twist. As applications, we obtain some examples of twists from Ocneanu cell calculus and Fendley–Ginsparg orbifold constructions. The relations between intertwiner and twist are also discussed, the groupoid structures are emphasized.