Double Lie structures in Poisson and symplectic geometry

The notion of a symplectic groupoid first arose as an apparatus to solve the quantization problem for Poisson manifolds. It is well known that not every Poisson manifold can be globally realised by a symplectic groupoid. A key motivation for this thesis was Lu and Weinstein's construction of a (global) symplectic double groupoid for an arbitrary Poisson Lie group. We develop an extensive exposition of their results, and analyse some of the possible extensions of their construction. In particular, we produce a symplectic double groupoid for any pair of dual Poisson groupoids where the underlying Lie groupoid structures are of trivial type. Alongside these ideas, we also study the actions of double Lie structures. A detailed account of the actions of double Lie groupoids is given, and notions for the actions of LA-groupoids are defined. As an application of these double actions, we consider an alternative approach to Xu's study of Poisson reduced spaces for actions of a symplectic groupoid. This approach is then extended to consider the Poisson reduced spaces for more general actions of Poisson groupoids.