The 3D Spin Geometry of the Quantum Two-Sphere

We study a three-dimensional differential calculus on the standard quantum two-sphere Sq2, coming from the 4D+ differential calculus on the quantum group SUq(2). We use a frame bundle approach to give an explicit description of Ω1Sq2 and its associated spin geometry in terms of a natural spectral triple over the sphere. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.