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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, which gives an interpretation of the quantum dynamical Yang–Baxter equation in terms of Hopf algebroids.