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Riemannian geometry of quantum groups and finite groups with nonuniversal differentials | | S. Majid
; | | Date: |
21 Jun 2000 | | Journal: | Commun.Math.Phys. 225 (2002) 131-170 | | Subject: | Quantum Algebra; Algebraic Geometry | math.QA gr-qc hep-th math.AG | | 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 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. | | Source: | arXiv, math.QA/0006150 | | Services: | Forum | Review | PDF | Favorites | |
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