Grothendieck Groups of Poisson Vector Bundles

Abstract

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Evidence is given that this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary Lie algebroids. Basic properties of the Poisson K-ring areproved and the Poisson K-rings are calculated for a number of examples. In particular, for the zero Poisson structure the K-ring is the ordinary K⁰-ring of the manifold and for the dual space to a Lie algebra the K-ring is the ring of virtual representations of the Lie algebra. It is also shown that the K-ring is an invariant of Morita equivalence. Moreover, the K-ring is a functor on a category, the weak morita category, which generalizes the notion of Morita equivalence of Poisson Manifolds.