In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra g = h ⊕ m, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix r : h* → Λ2g naturally corresponds to a Poisson manifold h* × G. A special type of quantization of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang-Baxter equation (or Gervais-Neveu-Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics