Podle\'s Spheres for the Braided Quantum $\operatorname{SU}(2)$

TL;DR

This paper constructs a quantum sphere model using braided quantum $ ext{SU}_q(2)$ and shows that the classification of quantum spaces with $ ext{SU}_q(2)$ action for complex $q$ aligns with Podleś's classification for real $q$, extending the theory.

Contribution

It extends Podleś's classification of quantum spaces to the braided quantum $ ext{SU}_q(2)$ with complex deformation parameter, establishing a correspondence with the real case.

Findings

Quantum sphere modeled as quotient of braided $ ext{SU}_q(2)$ by $ ext{T}$.

Classification of quantum spaces matches Podleś's for real $q$, now extended to complex $q$.

The spectral properties lead to the same family of quantum spaces for complex $q$ as in the real case.

Abstract

Starting with the braided quantum group $\operatorname{SU}_q(2)$ for a complex deformation parameter $q$ we perform the construction of the quotient $\operatorname{SU}_q(2)/\mathbb{T}$ which serves as a model of a quantum sphere. Then we follow the reasoning of Podle\'{s} who for real $q$ classified quantum spaces with the action of $\operatorname{SU}_q(2)$ with appropriate spectral properties. These properties can also be expressed in the context of the braided quantum $\operatorname{SU}_q(2)$ (with complex $q$) and we find that they lead to precisely the same family of quantum spaces as found by Podle\'{s} for the real parameter $|q|$.