Atiyah sequences of braided Lie algebras and their splittings

Given an equivariant noncommutative principal bundle, we construct an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. In the case of the principal SU.2/-bundle over the sphere S_θ⁴ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra so_θ .5; 1/ yields a five parameter family of splittings. On the principal SO_θ .2n; R/-bundle of orthonormal frames over the sphere S_θ²ⁿ, a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on S_θ²ⁿ. The corresponding Riemannian geometry of S_θ²ⁿ is worked out.