Grothendieck Groups of Poisson Vector Bundles

By V. Ginzburg. 2000

47 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/372a6cf5a775b06e19e8522202324de65bd051ff

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Generalized Kähler Geometry (2010, background): “Morita equivalence for Lie algebroids in the smooth category is well-studied in Poisson geometry [14,16,50] and the version we develop here is a special case, with additional refinements made possible by the complex structures which are present.”
Integrability of Lie brackets (2001, methodology): “This is explained in [5], [14], using the integrability of actions of [20], but it follows also from our construction of G(A) (see next section) since we have: Proposition 1.”
Lectures on integrability of Lie brackets (2006, unknown): “, [27, 32, 66, 69]) though computations are in general quite difficult.”