Cartan structure equations and Levi-Civita connection in noncommutative geometry

TL;DR

This paper develops a framework for differential and Riemannian geometry on noncommutative algebras with Hopf algebra symmetries, extending classical concepts like Cartan structure equations and Levi-Civita connections.

Contribution

It introduces a comprehensive approach to noncommutative geometry using braided modules, proving existence and uniqueness of Levi-Civita connections, and generalizing classical geometric identities.

Findings

Cartan structure equations are derived in the noncommutative setting.

Existence and uniqueness of Levi-Civita connections are established.

The framework includes Drinfeld twists and cotriangular Hopf algebras.

Abstract

We study the differential and Riemannian geometry of algebras $A$ endowed with an action of a triangular Hopf algebra $H$ and noncommutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a compact closed category of $H$-equivariant $A$-bimodules, whose internal morphisms correspond to tensor fields. Vector fields and forms approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) $A$-module (not necessarily $A$-bimodule) connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary pseudo-Riemannian metrics is proven via a Koszul formula. The general theory includes Drinfeld twists of commutative geometries and also cotriangular Hopf algebras. It is illustrated with the example of the tensor square of Sweedler Hopf algebra which becomes a noncommutative Einstein manifold via a non-central metric.