Riemannian geometry of quantum groups and finite groups with nonuniversal differentials

Details Riemannian geometry of quantum groups and finite groups with nonuniversal differentials Riemannian geometry of quantum groups and finite groups with nonuniversal differentials

2000-06-20

arxiv.org/abs/math/0006150v2

Commun.Math.Phys. 225 (2002) 131-170

Mathematics

Quantum Algebra

43 pages, 1 figure. Revised August 2001 to cut page length (eg deleted appendix) for publication in CMP. Also fleshed out ex.

Scientific paper

10.1007/s002201000564

We construct noncommutative `Riemannian manifold' structures on dual quasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariant differential calculus, using the quantum frame bundle formalism introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras $C[G]$ with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group $C[S_3]$ is worked out in full detail and a unique torsion free and cotorsion free or `Levi-Civita' connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as $S_3$. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form.

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