We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if ${}_{\Gamma }$ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space $k{\oplus }_{\Gamma }$ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra $U{(}_{\Gamma })$ is a bialgebra in the category of A-comodules.

It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podleś spheres are related by restriction. In this approach, the equatorial Podleś sphere is distinguished because only in this case the restricted spectral triple admits an equivariant grading operator together with a real structure (up to infinitesimals of arbitrary high order). The real structure is expressed by the Tomita operator on quantum SU(2) and it is...