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.
CITATION STYLE
Xu, P. (2001). Quantum groupoids. Communications in Mathematical Physics, 216(3), 539–581. https://doi.org/10.1007/s002200000334
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