## 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) |
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Pages (from-to) | 539-581 |

Number of pages | 43 |

Journal | Communications In Mathematical Physics |

Volume | 216 |

Issue number | 3 |

DOIs | |

State | Published - Feb 2001 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics