An algebraic framework for noncommutative bundles with homogeneous fibres

TL;DR

This paper introduces an algebraic framework for noncommutative bundles with quantum homogeneous fibers, utilizing principal coalgebra extensions to generalize principal bundles in noncommutative geometry, supported by concrete examples.

Contribution

It develops a new algebraic approach for noncommutative bundles with quantum homogeneous fibers using principal coalgebra extensions, including symmetry considerations.

Findings

Constructed noncommutative $ ext{CP}_q^1$-bundles as examples.

Described quantum flag and twistor bundles within the framework.

Demonstrated the applicability to quantum spheres and bundles.

Abstract

An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which might be additionally equipped with a Hopf algebra symmetry. The proposed framework is supported by two examples of noncommutative $\mathbb{C} P_q^1$-bundles: the quantum flag manifold viewed as a bundle with a generic Podle\'s sphere as a fibre, and the quantum twistor bundle viewed as a bundle over the quantum 4-sphere of Bonechi, Ciccoli and Tarlini.