Grothendieck groups of Poisson vector bundles (Q1872793)

Recently, Poisson geometry has become an active field of research. It was stimulated by connections with a number of areas, including harmonic analysis on Lie groups, infinite-dimensional Lie algebras, mechanics of particles and continua, singularity theory, and completely integrable systems, just to mention a few examples. Poisson manifolds have a wide variety of invariants, almost all of which are extremely difficult to compute. For example, Poisson (co)homology and its various derivatives are not calculated explicitly even for dual spaces to semisimple Lie algebras of non-compact type. In this paper, the author introduces a new invariant, a Poisson \(K\)-ring. It is shown that a Poisson \(K\)-ring is a more tractable invariant than a Poisson (co)homology. The author defines a version of this invariant for arbitrary Lie algebroids, proves basic properties of the Poisson \(K\)-rings, and calculates the Poisson \(K\)-rings 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. The author also shows that the Poisson \(K\)-ring is an invariant of Morita equivalence. Finally it is shown that the Poisson \(K\)-ring is a functor on a category, the weak Morita category that generalizes the notion of Morita equivalence of Poisson manifolds.