On Symplectic and Contact Groupoids

This paper deals with Lie groupoids, in particular symplectic and contact groupoids. We formulate a deenition of contact groupoids; we give examples and a counterexample (groupoid with a contact form which is not a contact groupoid). The theory of diierentiable groupoids (called now Lie groupoids) has been introduced by C. Ehresmann E] in 1950 in his paper on connections (where he deened the groupoid of paths, the groupoid associated with a principal bundle, the parallel displacement); then he utilized groupoids to deene Lie pseudogroups by means of jets; he deened prolongations of groupoids. His pupils, the author L], A. Haeeiger H], J. Pradines P] etc. worked on this subject. In particular, introducing the notion of Lie algebroid (which corresponds to Lie algebras in the case of Lie groups), J. Pradines proved the Lie third theorem (at least for local Lie groupoids). The works of I. Koll a r are connected with this subject Ko]. Utilizing diierent kinds of prolongations, Guillemin-Sternberg G-S], Kumpera-Spen-cer K-S] investigated the theory of Lie equations. Then appeared the work of K. Mackenzie M] on Lie groupoids and Lie algebroids in Diierential Geometry and the paper by J. Renault R] dealing with a groupoid approach to C algebras. Recently A. Weinstein W1] and independently M.V. Karasev Ka] had deened sym-plectic groupoids and they have shown their importance in the theory of Poisson man-ifolds. For instance, one of principal motivations to introduce this notion, for A. Wein-stein, was the quantization of Poisson manifolds. During the last ve years, the theory of symplectic groupoids has known a very large expansion with, among others, the works of A. Weinstein and his pupils, P. Dazord and his pupils (see W2], W3], W-X], C-D-W], D-S]). C. Albert and P. Dazord A-D] have given a geometric approach of symplectic groupoids. Our paper, which tries to be self-contained, could serve as an introduction to the above-mentioned publications. In particular it borrows ideas from the work of Albert-Dazord (slidings, deenition pseudogroup, deenition sheaf). This paper is in nal form and no version of it will be submitted for publication elsewhere.