Abstract

The group of symplectomorphisms of a symplectic manifold is a well-studied object. However, little is known about the Poisson diffeomorphism group of other Poisson structures. In this talk, we present a first step in this direction. First, we discuss the intricate algebraic structure of the Poisson diffeomorphism group and its connection with coisotropic bisections of a Poisson groupoid. Then, we reformulate the problem of finding smooth structures on these objects to a linearization problem of Poisson structures around Lagrangian submanifolds. In the second part, we dive deeper into linear Poisson structures and present a linearization result, generalizing the Lagrangian neighbourhood theorem to the setting of Lie algebroids and cosymplectic structures. This is applied to obtain smooth structures on Poisson diffeomorphism groups of several classes of Poisson manifolds. This is part of my master thesis, supervised by Ioan Marcut.