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.