Quantum symmetry of the toric noncommutative manifolds

In his framework, Rieffel showed that compact Lie groups of rank at least 2 admit nontrivial $\theta$-deformations as compact quantum groups. In my recent work, I showed that an action of such a Lie group $G$ on a manifold $M$ with a toric action can be extended to an action in the deformed setting. Of course, an action cannot be extended to an action of the deformed algebras for arbitrary $\theta$-parameters. First, I will explain what these deformations mean and I will explain exactly when an action in the classical setting extends to the noncommutative setting. I will also explain how the noncommutative 7-sphere $S^7_\theta$ can be viewed as a quantum homogeneous space.