The geometry of quantum lens spaces: real spectral triples and bundle structure

Abstract

We study almost real spectral triples on quantum lens spaces, as orbit spaces of free actions of cyclic groups on the spectral geometry on the quantum group $S{U}_q(2)$. These spectral triples are given by weakening some of the conditions of a real spectral triple. We classify the irreducible almost real spectral triples on quantum lens spaces and we study unitary equivalences of such quantum lens spaces. Applying a useful characterization of principal $U(1)$-fibrations in noncommutative geometry, we show that all such quantum lens spaces are principal $U(1)$-fibrations over quantum teardrops.

Revided according to referee comments. Statement and proof theorem 5.2 on principal bundles over quantum teardrops corrected. Accepted for publication in Mathematical Physics, Analysis and Geometry

References in corpus (4)

• On a Classification of Irreducible Almost-Commutative Geometries IV
• Pimsner algebras and Gysin sequences from principal circle actions
• Quantum weighted projective and lens spaces
• On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

Cited by in corpus (3)

• Lifting spectral triples to noncommutative principal bundles
• Pimsner algebras and circle bundles
• Gysin exact sequences for quantum weighted lens spaces