Quantum groupoids

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

Quantum dynamical Yang-Baxter equation over a nonabelian base

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra \frakg =\frakh \oplus \frakm, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix r: \frakh^* \lon \wedge^2 \frakg corresponds to a Poisson manifold \frakh^* \times G. A special type of quantizations of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang-Baxter equation (or Gervais-Neveu-Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed.Comment: 23 pages, minor changes made, final version to appear in Comm. Math. Phy