Quiver-theoretical approach to dynamical Yang–Baxter maps

A dynamical Yang–Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang–Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang–Baxter maps. Our key observation is that the category of dynamical sets over a set Λ introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang–Baxter maps, can be embedded into the category of quivers with vertices Λ. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang–Baxter maps and extend his construction to obtain a new class of dynamical Yang–Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang–Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.