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We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C[ SL], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for  an odd th root of unity that its eigenvalues are given by -integers [] for  = 0,1..., − 1 offset by the constant background curvature. We fully solve the Dirac equation for  = 3.

Noncommutative Ricci Curvature and Dirac Operator on Cq[SL2] at Roots of Unity

Noncommutative Ricci Curvature and Dirac Operator on C_q[SL₂] at Roots of Unity