Connes-Landi spheres are homogeneous spaces

TL;DR

This paper explores the structure of Connes-Landi spheres, showing they can be understood as homogeneous spaces through the lens of $ heta$-deformations of compact Lie groups and their associated quantum groups.

Contribution

It demonstrates that Connes-Landi spheres are homogeneous spaces by deriving them as fixed-point subalgebras of $ heta$-deformed compact Lie groups.

Findings

Connes-Landi spheres are realized as fixed-point subalgebras.

$ heta$-deformations produce noncommutative spheres with homogeneous space structure.

The approach links quantum group deformations to classical geometric structures.

Abstract

In this paper, we review some recent developments of compact quantum groups that arise as $\theta$-deformations of compact Lie groups of rank at least two. A $\theta$-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in \cite{W2018}, we derive the noncommutative 7-sphere in the sense of Connes and Landi \cite{CL2001} as the fixed-point subalgebra.