Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials |
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Authors: | Shahn Majid |
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Institution: | 1.School of Mathematical Sciences, Queen Mary, University of London, Mile End Road,?London E1 4NS, UK,GB |
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Abstract: | 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 approach 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
ℂG] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂS
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.
Received: 22 June 2000 / Accepted: 26 August 2001 |
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Keywords: | |
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