### Abstract

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* → Λ^{2}g 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.

Original language | English (US) |
---|---|

Pages (from-to) | 475-495 |

Number of pages | 21 |

Journal | Communications In Mathematical Physics |

Volume | 226 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2002 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics