Riemannian geometry of quantum groups and finite groups with nonuniversal differentials

Abstract

We construct noncommutative `Riemannian manifold' structures on dual quasitriangular Hopf algebras such as $C_q[S{U}_2]$ 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.

43 pages, 1 figure. Revised August 2001 to cut page length (eg deleted appendix) for publication in CMP. Also fleshed out ex. of q-Killing metric for q-SU_2 previously mentioned (no significant additions)

References in corpus (2)

• Conceptual Issues for Noncommutative Gravity on Algebras and Finite Sets
• $Z_2\times Z_2$ Lattice as a Connes-Lott-Quantum Group Model

Cited by in corpus (20)

• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Noncommutative geometry of angular momentum space U(su(2))
• Electromagnetism and Gauge Theory on the Permutation Group $S_3$
• Noncommutative Riemannian Geometry of the Alternating Group A_4
• Nonassociative Riemannian Geometry by Twisting
• Almost commutative Riemannian geometry: wave operators
• Universal $\kappa$-Poincaré covariant differential calculus over $\kappa$-Minkowski space
• Conceptual Issues for Noncommutative Gravity on Algebras and Finite Sets
• Noncommutative cohomology and electromagnetism on $C_q[S{L}_2]$ at roots of unity
• Differential Geometry of Group Lattices
• Involutive representations of coordinate algebras and quantum spaces
• Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry
• $Z_2\times Z_2$ Lattice as a Connes-Lott-Quantum Group Model
• Moduli of quantum Riemannian geometries on <= 4 points
• Warped Products and Yang-Mills equations on non commutative spaces
• Riemannian Geometry of Bicovariant Group Lattices
• Classification of differentials and Cartan calculus on bicrossproducts
• A Note on the Symmetries and Renormalisability of (Quantum) Gravity
• Poisson Principal Bundles
• Noncommutative geometry of the dihedral group D 6