Quivers with quantum Yang-Baxter equation and Hecke condition: Deformation of face algebras

In this paper we initiate the study of quivers carrying quantum Yang--Baxter and Hecke structure, and we apply this framework to study path algebras over quivers whose loop spaces carry RTT relations determined by Hecke $R$-matrices. We show that the quantum matrix algebra $\mathcal{O}_q(M_n)$ is isomorphic as a bialgebra to the face algebra over a rose quiver deformed by RTT relations of the $GL_q(n)$ Hecke $R$-matrix.