### 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 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Communications In Mathematical Physics*, vol. 226, no. 3, pp. 475-495. https://doi.org/10.1007/s002200200621

**Quantum dynamical Yang-Baxter equation over a nonabelian base.** / Xu, Ping.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Quantum dynamical Yang-Baxter equation over a nonabelian base

AU - Xu, Ping

PY - 2002/4/1

Y1 - 2002/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0036011684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036011684&partnerID=8YFLogxK

U2 - 10.1007/s002200200621

DO - 10.1007/s002200200621

M3 - Article

AN - SCOPUS:0036011684

VL - 226

SP - 475

EP - 495

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -