# Quiver-theoretical approach to dynamical Yang-Baxter maps

## TL;DR

This paper introduces a quiver-theoretical framework to analyze dynamical Yang-Baxter maps, extending Shibukawa's classification and connecting it with bialgebroids and weak bialgebras.

## Contribution

It develops a novel quiver-theoretical approach to dynamical Yang-Baxter maps, expanding classification and linking to algebraic structures.

## Key findings

- Embedded the category of dynamical sets into quivers with vertices b1
- Extended Shibukawa's classification of dynamical Yang-Baxter maps
- Connected dynamical Yang-Baxter maps with bialgebroids and weak bialgebras

## Abstract

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 $\Lambda$, 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 $\Lambda$. 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.