Higher symplectic structure on torsionless Lie-Rinehart pairs

Abstract

We define an n-plectic structure as a commutative and torsionless Lie Rinehart pair, together with a distinguished cocycle from its Chevalley-Eilenberg complex. This 'n-plectic cocycle' gives rise to an extension of the Chevalley-Eilenberg complex by so called symplectic tensors. The cohomology of this extension generalizes Hamiltonian functions and vector fields to tensors and cotensors in a range of degrees, up to certain coboundaries and has the structure of a Lie oo-algebra. Finally we show, that momentum maps appear in this context just as weak Lie oo-morphisms from an arbitrary Lie oo-algebra into the Lie oo-algebra of Hamiltonian (co)tensors.