Nonstandard Podles spheres as $U_c(\frak{h})$ invariants

Question

In this paper 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 $U_c({\frak h})$?

Answer 1

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)), Quantum groups, Proc. Workshops, Euler Int. Math. Inst. Leningrad/USSR 1990, Lect. Notes Math. 1510, 98-103 (1992). ZBL0743.33011.

[2] Dijkhuizen, Mathijs S.; Koornwinder, Tom H., Quantum homogeneous spaces, duality and quantum 2-spheres, Geom. Dedicata 52, No. 3, 291-315 (1994). ZBL0818.17017.