Quantization by Cochain Twists and Nonassociative Differentials

We show that several standard associative quantizations can be expressed as cochain module twists in the spirit of Moyal products, at least to O(~), but where the twist element is not a cocycle. This puts such quantisations induced by covariance groups into the setting of our previous work on semiclassicalisation of differential structures where we showed that that noncommutative geometry forces us into a hidden nonassociativity not visible in the algebra but visible in its differentials. The quantisations covered include: enveloping algebras U(g) as quantisations of g, a Fedosov-type quantisation of the sphere S under a Lorentz group covariance, the Mackey quantisation of homogeneous spaces, and the standard quantum groups Cq [G]. We also consider the differential quantisation of R for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the Dirac operator.