Yoichi Shibukawa : (H, X)-bialgebroids associated with dynamical Yang-Baxter maps and their dynamical representations

The R-matrix, a solution to the quantum Yang-Baxter equation, produces a quantum algebra through Faddeev-Reshetikhin-Sklyanin-Takhtajan (FRST for short) construction. Because this algebra is a bialgebra, its representations, also called L-operators for the R-matrix, form a tensor category. The main aim of this talk is to construct an algebra such that the category of its "representations" is isomorphic to the tensor category Rep R consisting of L-operators for a bijective dynamical Yang-Baxter map satisfying a weight zero condition. The FRST construction for this dynamical Yang-Baxter map implies an algebra called an (H, X)-bialgebroid, and it is the algebra. That is, we can define dynamical representations of this algebra, which form a tensor category that is isomorphic to Rep R. If time permits, we will explain categorical concepts, which are useful for the proof.