ON THE CLASSIFICATION OF QUANTUM LENS SPACES

There are many noncommutative deformations of classical spaces. For instance, the C*-algebras of quantum lens spaces can be defined as fixed-point subalgebras of the C*-algebras of Vaksman-Soibelman quantum spheres under actions of finite cyclic groups. Hong and Szymański described both the quantum spheres and quantum lens spaces, whose weights are all coprime to the order of the acting group, as graph C*-algebras. Later on, Brzeziński and Szymański generalised the result to include all quantum lens spaces. Unfortunately, as pointed out by Efren Ruiz, their description is not always correct. In this talk, I will first present a modified graph and sketch a proof that all quantum lens spaces are indeed graph C*-algebras. Then I will describe the classification results for graph C*-algebras of finite graphs by Eilers, Restorff, Ruiz and Sørensen in the setting of type-I graph C*-algebras. As an application of their classification result, they classified quantum lens spaces of dimension 7 whose weights are all coprime to the order of the acting group. In my recent joint work with Thomas Gotfredsen, we further investigated this classification and obtained a number-theoretic invariant for quantum lens spaces of dimension 7 where all but a single weight are coprime to the order of the acting group.