2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64336We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, 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 $q$ an odd $r$'th root of unity that its eigenvalues are given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$.16 pages amslatex. Minor revision to Dirac operator, now solving fully for $r=3$Quantum AlgebraHigh Energy Physics - TheoryDifferential Geometry58B32, 58B34text