Connes-Landi spheres are homogeneous spaces

In this paper, we review some recent developments of compact quantum groups that arise as the noncommutative toric deformation of compact Lie groups of rank at least two. Noncommutative toric deformation is merely a 2-cocycle deformation using an action of an \(n\)-torus of higher dimension than 2. Using the formula (Lemma 5.3) developed in \cite{W2018}, we reconstruct the Connes-Landi noncommutative 7-sphere as a homogeneous space. This is achieved by considering the coaction of the deformation of the coaction of special unitary groups.