Tangent Dirac structures

Abstract

The lift of a closed 2-form Omega on a manifold Q to a closed 2-form on TQ may be achieved by pulling back the canonical symplectic structure on T*Q by the bundle map Omega : TQ to T*Q. It is also known how to lift a Poisson structure on Q to a Poisson structure on TQ. The author calls these lifted structures 'tangent' structures. The notion of a Dirac structure is reviewed. This is a hybrid of Poisson and pre-symplectic structures, which may be thought of as a (singular) foliation of Q by pre-symplectic leaves. The main result of this paper is a single method which achieves the lift to TQ of either a Poisson or a pre-symplectic structure on Q. Natural involution on TTQ is shown to preserve the lift to TTQ of the lifted structure on TQ. The method is then applied to Dirac structures, with the result that a Dirac structure on Q has a tangent lift to TQ generalizing the lifts of Poisson and pre-symplectic structures.