Quantum Dynamical Ř- Matrix with Spectral Parameter from Fusion

Since the classical dynamical r-matrix [1] first appeared on the scene of integrable many body system, many dynamical r-matrices have been found in integrable models such as CalogeroMoser model [2], Sine-Gorden soliton case [3] and the general case for the Ruijsenaars systems [4]. These dynamical r-matrices do not satisfy the ordinary classical Yang-Baxter equation, so its quantization is rather nontrivial. The quantum dynamical Yang-Baxter (QDYB) equation, which appeared first in the quantization of Toda field theory [5] and later in the quantization of KZB equation [6], had been studied widely for various integrable models and its algebraic structure was explored [7, 8, 9]. In contrast to the non-dynamical one [10], only a few dynamical R matrices are constructed explicitly and most of them can be obtained from Felder’s solution [6] by taking a gauge transformations [9]. So how to construct new R matrix is still an interesting and challenging problem. As an efficient method to obtain higher-spin R matrix, fusion procedure [11] has been applied to dynamical R matrix [12]. In this paper, we construct a spin-1 quantum dynamical R-matrix with spectral parameter by ”fusing” together the spin1 2 R-matrices which satisfy the QDYB equation [7]: