Poisson–Riemannian geometry

Abstract

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter λ , using a functorial approach. This leads us to field equations of ‘Poisson–Riemannian geometry’ between the classical metric, the Poisson bracket and a certain Poisson-compatible connection needed as initial data for the quantisation of the differential structure. We use such data to define a functor Q to O ( λ 2 ) from the monoidal category of all classical vector bundles equipped with connections to the monoidal category of bimodules equipped with bimodule connections over the quantised algebra. This is used to ‘semiquantise’ the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection in the sense of constructing a noncommutative geometry to O ( λ 2 ) . We solve our field equations for the Schwarzschild black-hole metric under the assumption of spherical symmetry and classical dimension, finding a unique solution and the necessity of nonassociativity at order λ 2 , which is similar to previous results for quantum groups. The paper also includes a nonassociative hyperboloid, nonassociative fuzzy sphere and our previously algebraic bicrossproduct model.