Podleś Spheres for the Braided Quantum $\mathrm{SU}(2)$

Abstract

Starting with the braided quantum group ${\mathrm{SU}}_q(2)$ for a complex deformation parameter $q$ we perform the construction of the quotient ${\mathrm{SU}}_q(2)/\mathbb{T}$ which serves as a model of a quantum sphere. Then we follow the reasoning of Podleś who for real $q$ classified quantum spaces with the action of ${\mathrm{SU}}_q(2)$ with appropriate spectral properties. These properties can also be expressed in the context of the braided quantum ${\mathrm{SU}}_q(2)$ (with complex $q$) and we find that they lead to precisely the same family of quantum spaces as found by Podleś for the real parameter $|q|$.

25 pages (19 pages + appendix)