Poisson-Riemannian geometry

JOURNAL OF GEOMETRY AND PHYSICS

卷:114

Article

Beggs, Edwin J.¹  Majid, Shahn² 

[1] Swansea Univ, Dept Math, Singleton Parc, Swansea SA2 8PP, W Glam, Wales

[2] Queen Mary Univ London, Sch Math Sci, Mile End Rd, London EL 4NS, England

关键词: Noncommutative geometry;    Poisson geometry;    Quantum groups;    Quantum gravity;    Symplectic connection;    Monoidal functor;   

DOI  :  10.1016/j.geomphys.2016.12.012

来源: Elsevier

【 摘 要 】

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter lambda, 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 0(lambda(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(lambda(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 lambda(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. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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