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

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Abstract

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C q [ 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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  1. School of Mathematical Sciences, Queen Mary, University of London, Mile End Rd, London, E1 4NS, U.K.

    Shahn Majid

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  1. Shahn Majid
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Majid, S. Noncommutative Ricci Curvature and Dirac Operator on C q [SL2] at Roots of Unity. Letters in Mathematical Physics 63, 39–54 (2003). https://doi.org/10.1023/A:1022980227093

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  • DOI: https://doi.org/10.1023/A:1022980227093

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