Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

Details Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

2002-06-19

arxiv.org/abs/math/0206187v2

Mathematics

Quantum Algebra

16 pages amslatex. Minor revision to Dirac operator, now solving fully for $r=3$

Scientific paper

We 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 $r=3$.

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