Restricting the bi-equivariant spectral triple on quantum SU(2) to the Podles spheres

It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podles spheres are related by restriction. In this approach, the equatorial Podles 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 shown that the failure of the real structure to satisfy the commutant property is related to the failure of the universal R-matrix operator to be unitary.