The universal covering and covered spaces of a symplectic Lie algebra action Juan-Pablo Ortega 1 and Tudor S. Ratiu 2 Dedicated to Alan Weinstein on the occasion of his 60th birthday. The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein , J.E. Marsden and T.S. Ratiu, eds.), Progress in Mathematics 232, Birkh¨ auser-Verlag, Boston, 2004, 571–581. Abstract We show that the category of Hamiltonian covering spaces of a given connected and paracompact symplectic manifold (M,ω) acted canonically upon by a Lie algebra admits a universal covering and covered space. 1 Introduction Let (M,ω) be a connected symplectic manifold and g be a Lie algebra acting symplectically on it. A Lie algebra action of g on M is a Lie algebra antihomomorphism ξ ∈ g → ξ M ∈ X(M ) such that the map (m, ξ ) ∈ M × g ∗ → ξ M (m) ∈ TM is smooth. The action is symplectic when £ ξ M ω = 0, for any ξ ∈ g and where £ ξ M is the Lie derivative operator defined by the vector field ξ M . Definition 1.1 Let (M,ω) be a connected symplectic manifold and g be a Lie algebra acting symplec- tically on it. We say that the map p N : N → M is a Hamiltonian covering map of (M,ω) when it satisfies the following conditions: (i) p N is a smooth covering map (ii) (N,ω N ) is a connected symplectic manifold (iii) p N is a symplectic map 1 Centre National de la Recherche Scientifique, D´ epartement de Math´ ematiques de Besan¸ con, Universit´ e de Franche-Comt´ e, UFR des Sciences et Techniques. 16, route de Gray. F-25030 Besan¸ con cedex. France. Juan- Pablo.Ortega@math.univ-fcomte.fr 2 Institut de Math´ ematiques Bernoulli, ´ Ecole Polytechnique F´ ed´ erale de Lausanne, CH-1015 Lausanne, Switzerland. Tudor.Ratiu@epfl.ch 1