Quantum groupoids

Details Quantum groupoids Quantum groupoids

1999-05-30

arxiv.org/abs/math/9905192v2

Commun.Math.Phys. 216 (2001) 539-581

Mathematics

Quantum Algebra

48 pages, typos and minor mistakes corrected, references updadted. Comm. Math. Physics, (to appear)

Scientific paper

10.1007/s002200000334

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 ${\cal D}\otimes_{\hbar} U_{\hbar}(\frakg)$, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

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