### Abstract

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid D⊗ℏUℏg, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

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

Pages (from-to) | 539-581 |

Number of pages | 43 |

Journal | Communications In Mathematical Physics |

Volume | 216 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1 2001 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*216*(3), 539-581. https://doi.org/10.1007/s002200000334

}

*Communications In Mathematical Physics*, vol. 216, no. 3, pp. 539-581. https://doi.org/10.1007/s002200000334

**Quantum groupoids.** / Xu, Ping.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Quantum groupoids

AU - Xu, Ping

PY - 2001/2/1

Y1 - 2001/2/1

N2 - We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid D⊗ℏUℏg, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

AB - We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid D⊗ℏUℏg, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

UR - http://www.scopus.com/inward/record.url?scp=0035529585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035529585&partnerID=8YFLogxK

U2 - 10.1007/s002200000334

DO - 10.1007/s002200000334

M3 - Article

AN - SCOPUS:0035529585

VL - 216

SP - 539

EP - 581

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -