Quantum groupoids

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100 Scopus citations


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 languageEnglish (US)
Pages (from-to)539-581
Number of pages43
JournalCommunications In Mathematical Physics
Issue number3
StatePublished - Feb 2001

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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