On the group of Lagrangian bisections of a symplectic groupoid (Q2782574)

A symplectic groupoid \((\Gamma ,\omega)\) is a Lie groupoid \(\Gamma\) over \(\Gamma _0\) (the manifold of units) endowed with a symplectic structure \(\omega\) compatible in a certain way. This notion extends the concept of symplectomorphism group. A first important property is that, for a symplectic groupoid, there exists an induced Poisson structure \(\Lambda\) on \(\Gamma _0\). NEWLINENEWLINENEWLINEIn this paper, it is shown that the group of Lagrangian bisections of a symplectic groupoid is a regular (infinite dimensional) Lie group in the convenient setting of \textit{A. Kriegl} and \textit{P. W. Michor} [Mathematical Surveys and Monographs 53, Am. Math. Soc. (1997; Zbl 0889.58001)]. NEWLINENEWLINENEWLINEMoreover, the author extends the concept of flux homomorphism for the group of Lagrangian bisections. The flux is also obtained in its local form, which is used in order to characterize exact isotopies. NEWLINENEWLINENEWLINERecently, M. Crainic and R.L. Fernandes have obtained, using their general results about the integration of Lie algebroids (the infinitesimal invariants of Lie groupoids), necessary and sufficient conditions for a Poisson manifold to be integrable to a symplectic groupoid [Integrability of Poisson brackets, arXiv: math.DG/0210152].NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].