We semiclassicalise the
standard notion of differential calculus in noncommutative geometry on
algebras and quantum groups. We show in the symplectic case that the
infinitesimal data for a differential calculus is a symplectic connection, and
interpret its curvature as lowest order nonassociativity of the exterior algebra.
Semiclassicalisation of the noncommutative torus provides an example with zero
curvature. In the Poisson–Lie group case we study left-covariant infinitesimal data in
terms of partially defined preconnections. We show that the moduli space of
bicovariant infinitesimal data for quasitriangular Poisson–Lie groups has a
canonical reference point which is flat in the triangular case. Using a theorem of
Kostant, we completely determine the moduli space when the Lie algebra
is simple: the canonical preconnection is the unique point other than in
the case of sln, n > 2, when the moduli space is 1-dimensional. We relate
the canonical preconnection to Drinfeld twists and thereby quantise it to a
super coquasi-Hopf exterior algebra. We also discuss links with Fedosov
quantisation.
