Noncommutative Fibre Bundles via Bimodules

TL;DR

This paper develops a spectral sequence framework for noncommutative fibre bundles using bimodules with zero-curvature connections, extending classical differential geometry concepts to noncommutative algebras.

Contribution

It introduces a novel spectral sequence construction for noncommutative fibre bundles employing bimodule connections, generalizing differentiable algebra maps to positive maps via KSGNS.

Findings

Constructed a Leray-Serre spectral sequence for noncommutative de Rham cohomology.

Provided examples involving group algebras, matrix algebras, and quantum tori.

Extended classical geometric notions to noncommutative settings.

Abstract

We construct a Leray-Serre spectral sequence for fibre bundles for de Rham sheaf cohomology on noncommutative algebras. The morphisms are bimodules with zero-curvature extendable bimodule connections. This generalises definitions involving differentiable algebra maps to differentiable completely positive maps by using the KSGNS construction and Hilbert $C^*$-bimodules with bimodule connections. We give examples of noncommutative fibre bundles, involving group algebras, matrix algebras, and the quantum torus.