Atiyah sequences of braided Lie algebras and their splittings

Associated with an equivariant noncommutative principal bundle we give an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. For the SU(2)-principal bundle over the quantum 4-sphere an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra yields a five parameter family of splittings. On the principal bundle of orthonormal frames over the 2n-dimensional quantum sphere, the splitting of the sequence leads to the Levi-Civita connection for the `round' metric. The corresponding Riemannian geometry is worked out.