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# Journal of Symplectic Geometry

## Volume 1 (2001)

### Number 1

### Grothendieck Groups of Poisson Vector Bundles

Pages: 121 – 170

DOI: http://dx.doi.org/10.4310/JSG.2001.v1.n1.a4

#### Author

#### 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 K^{0}-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.