Locally vacant double Lie groupoids and the integration of matched pairs of Lie algebroids (Q1125242)

Two types of integration resultS for Poisson Lie groups have been considered by many authors. As representative see \textit{J.-H. Lu} and \textit{A. Weinstein} [C. R. Acad. Sci., Paris, Sér. I 309, No. 18, 951-954 (1989; Zbl 0701.58025)] and \textit{S. Majid} [Pac. J. Math. 141, No. 2, 311-332 (1990; Zbl 0735.17017)] respectively. In [Duke Math. J. 86, No. 2, 261-304 (1997; Zbl 0889.58036)] \textit{J.-H. Lu} has shown that any Poisson action give rise to a match pair of Lie algebroids and the global structure underlying a Poisson action is a matched pair of Lie groupoids. The authors proved in previous works that matches pairs of Lie groupoids have as differential correspondents \(\mathcal{LA}\)-groupoids [the first author, (*) Adv. Math. 94, No. 2, 180-239 (1992; Zbl 0765.57025)] and the matches pairs of Lie algebroids [the second author, Glasg. Math. J. 39, No. 2, 167-181 (1997; Zbl 0886.22012)]. Using the theory of double Lie groupoids, as developed by the first author in [(*)], in the paper is given a general integrability result for matched pairs of Lie algebroids, which combine the previous approaches. A new class of double Lie groupoids called locally vacant, which satisfy an étale form of the vacancy condition, is introduced. Using an analogous method of Lu and Weinstein [loc. cit.], applied to a general Lie groupoid, one yields to a double groupoid of this type and one calculates the \(\mathcal{LA}\)-groupoid. One proves that a locally vacant double Lie groupoid induces a matches pair of Lie algebroids. The main result states that a diagonally integrable [see the authors in Glasg. Math., loc. cit.]) match pair of Lie algebroids integrate to a locally vacant Lie groupoid.