Grothendieck Groups of Poisson Vector Bundles

Details Grothendieck Groups of Poisson Vector Bundles Grothendieck Groups of Poisson Vector Bundles

2000-09-13

arxiv.org/abs/math/0009124v1

Mathematics

Differential Geometry

32 pages

Scientific paper

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved 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-theory 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.

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