*-Compatible Connections in Noncommutative Riemannian Geometry

Abstract

We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that *-compatible bimodule connections lead to braid operators $σ$ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert *-modules and metric compatibility in both the usual and a cotorsion form. We show that the theory works well for the quantum group $C_q[SU_2]$ with its 3D calculus, finding for each point of a 3-parameter space of covariant metrics a unique `Levi-Civita' connection deforming the classical one and characterised by zero torsion, metric-preservation and *-compatibility. Allowing torsion, we find a unique connection with classical limit that is metric-preserving and *-compatible and for which $σ$ obeys the braid relations. It projects to a unique `Levi-Civita' connection on the quantum sphere. The theory also works for finite groups and in particular for the permutation group $S_3$ where we find somewhat similar results.