In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle E → M , consists of an antisymmetric bracket on the sections of E whose "Jacobi anomaly" has an explicit expression in terms of a bundle map E → T M and a field of symmetric bilinear forms on E. When M is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate symmetric bilinear form.For any Lie bialgebroid (A, A * ) over M (a notion defined by Mackenzie and Xu), there is a natural Courant algebroid structure on A ⊕ A * which is the Drinfel'd double of a Lie bialgebra when M is a point. Conversely, if A and A * are complementary isotropic subbundles of a Courant algebroid E, closed under the bracket (such a bundle, with dimension half that of E, is called a Dirac structure), there is a natural Lie bialgebroid structure on (A, A * ) whose double * 1991 Mathematics Subject Classification. Primary 58F05. Secondary 17B66, 22A22, 53C99, 58H05.

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We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincaré-BirkhoffWitt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S 1 -bundles and S 1 -gerbes over differentiable stacks. In particular, we establish the relationship between S 1 -gerbes and groupoid S 1 -central extensions. We define connections and curvings for groupoid S 1 -central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S 1 -gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S 1 -bundles and S 1 -gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S 1 -central extensions with prescribed curvature-like data.2 folklore (see [15,34,39,40]). However, we feel that it is useful to spell it out in detail in the differentiable geometry setting, which is of ultimate interest for our purpose.Our main goal of this paper is to develop the theory of S 1 -gerbes over differentiable stacks. Motivation comes from string theory in which "gerbes with connections" appear naturally [13,16,23,46]. For S 1 -gerbes over manifolds, there has been extensive work on this subject pioneered by Brylinski [5], Chatterjee [8], Hitchin [21], Murray [32] and many others. Also, there is interesting work on equivariant S 1 -gerbes, e.g., by Brylinski [6], Meinrenken [29], Gawedzki-Neis [17], Stienon [41] and others, as well as on gerbes over orbifolds [27]. These endeavors make the foundations of gerbes over differentiable stacks a very important subject. An important step is to geometrically realize a class H 2 (X, S 1 ) (or H 3 (X, Z) when X is Hausdorff). Such a geometrical realization is crucial in applications to twisted K-theory [43,44,45].Our method is to use the dictionary mentioned above, under which we show that S 1 -gerbes are in one-to-one correspondence with Morita equivalence classes of groupoid S 1 -central extensions. Thus it follows from a well-known theorem of Giraud [19] that there is a bijection between H 2 (X, S 1 ) and Morita equivalence classes of Lie groupoid S 1 -central extensions. We note that there are several independent investigations of similar topics; see [7,36,37,42,50].An S 1 -central extension of a Lie groupoid X 1 ⇉ X 0 is a Lie groupoid R 1 ⇉ X 0 with a groupoid morphism π : R 1 → X 1 such that ker π ∼ = X 0 × S 1 lies in the center of R 1 . It is easy to see that π : R 1 → X 1 is then naturally an S 1 -principal bundle. A standard example is an S 1 -central extension of aČech groupoid: Let N be a manifold and α ∈ H 3 (N, Z), and let {U i } be a good covering of N . Then the groupoidwhich is called theČech groupoid, is Morita equivalent to the manifold N . Then the S 1 -gerbe corresponding to the class α can be realized as anare the same point x in the three-intersection U ijk considered as elements in the two-intersections, and c ijk : U ijk → S ...

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⊗ U g, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.