Principal fibrations over noncommutative spheres

TL;DR

This paper constructs noncommutative four-spheres as base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces, introducing new examples and analyzing their geometric properties.

Contribution

It provides explicit examples of noncommutative four-spheres with principal bundle structures over noncommutative spheres, including conditions on their Connes--Chern characters.

Findings

Noncommutative four-spheres constructed as base spaces.

Existence of $SU(2)$-principal bundles with noncommutative seven-spheres.

Identification of a non-zero Hochschild cycle acting as a volume form.

Abstract

We present examples of noncommutative four-spheres that are base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of $SU(2)$. We give conditions for the components of the Connes--Chern character of the projection to vanish but the second (the top) one. The latter is then a non zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres.