Lifts of Poisson and Related Structures

The derivation d T on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle T M is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups. 0. Introduction. A derivation d T on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms of the tangent bundle TM plays essential rôle in the calculus of variations ([Tu]) and, in particular, in analytical mechanics. The derivation d T ω of the symplectic 2-form of a symplectic manifold (M, ω) provides the tangent bundle TM with a symplectic structure. A vector field X: M → TM is locally Hamiltonian if its image X(M) is a Lagrangian submanifold of (TM, d T ω). The concept of a generalized hamiltonian system can be introduced as a Lagrangian submanifold of (TM, d T ω). The infinitesimal dynamics of a relativistic particle is an example of such a system. The derivation d T has also an aspect of the total Lie derivative in the exterior algebra of forms: £ X µ = X * d T µ (Theorem 3.2). In analytical mechanics Poisson structures play the role as important as symplectic structures. The phase space is considered as a manifold equipped with a Poisson structure rather than symplectic one. On the other hand, in the theory of systems with symmetries, much attention is paid to the case of Poisson symmetries, i. e., the symmetry group is a Poisson-Lie group. Poisson-Lie groups are of interest also because of their relation to quantum groups. A Poisson structure is usually given by a bivector field Λ and, in general, not by a two-form and vanishing of the Schouten bracket corresponds to vanishing of the exterior derivative. This shows that, in order to generalize the mentioned ideas and results from the symplectic to the Poisson case, we need to carry over the discussion from forms to multivector fields. The aim of this paper is to extend d T to the exterior algebra of multi-vector fields and to establish relations concerning Poisson structures which correspond to the mentioned above relations in symplectic geometry. We would like to emphasize that our goal is not to extend the general theory of derivations of forms or vector-valued forms …