Atiyah sequences of braided Lie algebras and their splittings

Abstract

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 $\mathrm{SU}(2)$-bundle over the sphere $S_{\theta }^4$ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra ${\mathrm{so}}_{\theta }(5,1)$ yields a five parameter family of splittings. On the principal ${\mathrm{SO}}_{\theta }(2n,\mathbb{R})$-bundle of orthonormal frames over the sphere $S_{\theta }^{2n}$, a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on $S_{\theta }^{2n}$. The corresponding Riemannian geometry of $S_{\theta }^{2n}$ is worked out.