Noncommutative Ricci Curvature and Dirac Operator on C q [SL2] at Roots of Unity |
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Authors: | Majid Shahn |
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Affiliation: | (1) School of Mathematical Sciences, Queen Mary, University of London, Mile End Rd, London, E1 4NS, U.K. |
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Abstract: | We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group Cq[ SL2], 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 rth 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. |
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Keywords: | gravity Jones index noncommutative geometry quantum groups Riemann tensor roots of unity spinor |
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