Grothendieck Groups of Poisson Vector Bundles

TL;DR

This paper introduces a new Poisson K-ring invariant for Poisson manifolds, which generalizes classical K-theory and is invariant under Morita equivalence, with calculations for key examples.

Contribution

It defines the Poisson K-ring invariant, explores its properties, and demonstrates its invariance under Morita equivalence, extending K-theory to Poisson geometry.

Findings

Poisson K-ring generalizes classical K-theory for Poisson manifolds.

The K-ring is invariant under Morita equivalence.

Explicit calculations of the K-ring for key examples.

Abstract

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.