The isospectral Dirac operator on the 4-dimensional orthogonal quantum sphere

Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere Sq4. These representations are the constituents of a spectral triple on Sq4 with a Dirac operator which is isospectral to the canonical one on the round sphere S4 and which then gives 4+ summability. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an ‘instanton’ projection. We also introduce a real structure which satisfies all required properties modulo smoothing operators.