The affine group as symplectic manifold (Q1313230)

Let \((G,\Omega)\) be a symplectic Lie group i.e. a Lie group carring a left invariant symplectic form \(\Omega\). Let \({\mathfrak L}\) be a left invariant Lagrangian foliation on \((G,\Omega)\) and \(H\) the leaf through \(\varepsilon\) \((\varepsilon\) the neutral element of \(G)\). If the product action \(H\times G\to G\) is a Hamiltonian action, it is shown that the leaves of \({\mathfrak L}\) are closed. In particular the paper studies the left invariant symplectic structures \((\Omega=d\nu)\) on the affine group \(GA(\mathbb{K}^ n)=\mathbb{K}^ n\rtimes GL(\mathbb{K}^ n)\) for \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\). The work proves that \(GA(\mathbb{K}^ n)\) is endowed with: i) a left invariant Lagrangian foliation \({\mathfrak L}\) where the associated vector bundle \(L\subset TG\) is included in the kernel of \(\nu\), ii) two transversal left invariant symplectic foliations. Furthermore it is shown that for a certain left invariant symplectic structure of \(GA(\mathbb{K}^ n)\) it exists an exact symplectic embedding of \(G\) onto an open subset of the cotangent bundle of the Euclidean group \(\mathbb{R}^ n\rtimes SO(\mathbb{R}^ n)\).