Noncommutative vector field calculi

We discuss noncommutative differential geometry from a vector field centric point of view.

This is based on the notion of first order vector field calculus (FOVC), which has been previously introduced by Borowiec under the name Cartan pair.

We define the universal FOVC and construct an adjunction between the categories of FOVC and that of first order differential calculi, showing that the vector field approach is dual, though not equivalent, to the differential form one.

This correspondence is then extended to covariant vector field and differential calculi.

On Hopf algebras, (bi)covariant FOVC are in bijection with (bicovariant) quantum tangent spaces.

For Hopf--Galois extensions, quantum tangent spaces give rise to vertical vector fields and in this setup we further describe base vector fields and horizontal vector fields and show that they are related via a noncommutative Atiyah sequence.

Multiple examples, based on braided derivations, finite groups and the quantum Hopf fibration, are given.

The vector field approach is further enriched by a sheaf-theoretic treatment, which recovers the former as a local, or affine, picture.