In this [paper](https://link.springer.com/content/pdf/10.1007/BF00416848.pdf) Podles introduced a $2$-parameter family of $q$-deformed spheres $S\_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "standard Podles sphere". All algebras can be realised as subalgebras of the quantum group $SU\_q(2)$ and for this case $c=0$ we have a well known description of $S\_{q,0}$ as the space of invariants of $U({\frak h})$ the Hopf-subalgebra of $U\_q({\frak sl}\_2)$ generated by the elements $K$ and $K^{-1}$. What happens for the other cases? Is there a general family of algebras $U\_c({\frak h})$ such that $S\_{q,c}$ is the space of invariants of $U\_c({\frak h})$?



Nonstandard Podles spheres as $U_c(\frak{h})$ invariants
Yes, there is a one parameter family of coideal subalgebra of $U\_q(\mathfrak{sl}\_2)$ that give the Podleś sphere algebras as their coinvariants. The generators of those coideals are given in [1]. More conceptually, there is a duality between the coideals of "function algebra" and those of "universal enveloping algebra" of quantum groups [2].


[1] *Noumi, Masatoshi; Mimachi, Katsuhisa*, [**Askey-Wilson polynomials as spherical functions on (\mathrm{SU}\_q(2))**](http://dx.doi.org/10.1007/BFb0101182), Quantum groups, Proc. Workshops, Euler Int. Math. Inst. Leningrad/USSR 1990, Lect. Notes Math. 1510, 98-103 (1992). [ZBL0743.33011](https://zbmath.org/?q=an:0743.33011).


[2] *Dijkhuizen, Mathijs S.; Koornwinder, Tom H.*, [**Quantum homogeneous spaces, duality and quantum 2-spheres**](http://dx.doi.org/10.1007/BF01278478), Geom. Dedicata 52, No. 3, 291-315 (1994). [ZBL0818.17017](https://zbmath.org/?q=an:0818.17017).


