Quantization by cochain twists and nonassociative differentials

We show that several standard associative quantizations in mathematical physics can be expressed as cochain module-algebra twists in the spirit of Moyal products at least to O(ℏ3), but to achieve this we twist not by a 2-cocycle but by a 2-cochain. This implies a hidden nonassociativity not visible in the algebra itself but present in its deeper noncommutative differential geometry, a phenomenon first seen in our previous work on semiclassicalization of differential structures. The quantizations are induced by a classical group covariance and include enveloping algebras U(g) as quantizations of g∗, a Fedosov-type quantization of the sphere S2 under a Lorentz group covariance, the Mackey quantization of homogeneous spaces, and the standard quantum groups Cq[G]. We also consider the differential quantization of Rn for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the Dirac operator.