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A [[C*-algebra|$ C^{*} $-algebra]] $ \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) $ generated by two elements $ A $ and $ B $ satisfying [[#References|[a1]]]
\begin{gather}
A^{*} = A, \qquad
B A = q^{2} A B, \\
B^{*} B = A - A^{2} + c \mathbf{1}, \qquad
B B^{*} = q^{2} A - q^{4} A^{2} + c \mathbf{1}.
\end{gather}
Here, $ q \in \Bbb{R} $ is a deformation parameter and $ c \in \Bbb{R} $ is another parameter labeling the family of quantum spheres. Each quantum sphere is a [[Quantum homogeneous space|quantum homogeneous space]] in the sense that there exists a right co-action
$$
\text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) \to \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) \otimes {\text{Fun}_{q}}(\text{SU}(2)),
$$
where $ {\text{Fun}_{q}}(\text{SU}(2)) $ stands for the quantum $ \text{SU}(2) $-group [[#References|[a7]]] (cf. also [[Quantum groups|Quantum groups]]) considered as a deformation of the [[Poisson algebra|Poisson algebra]] $ \text{Fun}(\text{SU}(2)) $. The one-parameter family of quantum spheres is in correspondence with the family of $ \text{SU}(2) $-covariant Poisson structures on $ \Bbb{S}^{2} $, which is known to be one-parametric too ([[#References|[a2]]], Appendix). The deformation of the Poisson structure $ \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) $ can be introduced in a precisely defined manner [[#References|[a2]]]. Also, the structure of representations of $ \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) $ is in correspondence with the structure of symplectic leaves on $ \Bbb{S}^{2}_{c} $ [[#References|[a1]]], [[#References|[a2]]]. For $ c &gt; 0 $, the symplectic leaves are two open discs and the points of a circle separating them. For $ c = 0 $, one disc leaf is attached to one one-point leaf and, in fact, this is the Bruhat decomposition of the Poisson homogeneous space $ \text{U}(1) \setminus \text{SU}(2) $. For $ c &lt; 0 $, $ \Bbb{S}^{2}_{c} $ is a [[Symplectic manifold|symplectic manifold]].

The symplectic spheres $ \Bbb{S}^{2}_{c} $, with $ c &lt; 0 $, can be realized as orbits of the dressing transformation of $ \text{SU}(2) $ acting on its dual [[Poisson Lie group|Poisson Lie group]]. An equivalent realization is given by the right $ \text{SU}(2) $-action on the manifold $ M $ of $ (2 \times 2) $-unimodular positive matrices, which is just the unitary transformation $ (m,u) \mapsto u^{*} m u $. There exists a quantum analogue as a right co-action $ {\text{Fun}_{q}}(M) \to {\text{Fun}_{q}}(M) \otimes {\text{Fun}_{q}}(\text{SU}(2)) $, which is defined formally in the same way as in the Poisson case. If $ c(n) = - \dfrac{q^{2 n}}{(1 + q^{2 n})^{2}} $, $ n = 1,2,\ldots $, one can construct, using this structure, the $ n $-dimensional irreducible representation of the deformed universal enveloping algebra $ {\mathcal{U}_{q}}(\mathfrak{su}(2)) $ [[#References|[a3]]]. Moreover, if $ c &lt; 0 $, then the $ C^{*} $-algebra $ \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) $ is non-trivial only for $ c = c(n) $ [[#References|[a1]]].

A few other concepts have been developed for quantum spheres, including a description in terms of a local holomorphic coordinate $z$ and its adjoint $ z^{*} $ [[#References|[a3]]], [[#References|[a4]]] and a differential and integral calculus [[#References|[a4]]], [[#References|[a5]]]. In a precise analogy with the classical case, quantum spherical functions were defined as special basis elements in
$$
\text{Fun}_{q} \left( \Bbb{S}^{2}_{c = 0} \right) \equiv \text{Fun}_{q}(\text{U}(1) \setminus \text{SU}(2))
$$
and expressed in terms of big $ q $-Jacobi polynomials [[#References|[a6]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top"> P. Podles, “Quantum spheres” ''Lett. Math. Phys.'', '''14''' (1987) pp. 193–202.</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top"> A.J.-L. Sheu, “Quantization of the Poisson $ \text{SU}(2) $ and its Poisson homogeneous space — the $ 2 $-sphere” ''Comm. Math. Phys.'', '''135''' (1991) pp. 217–232.</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top"> P. Šťovíček, “Quantum line bundles on $ S^{2} $ and the method of orbits for $ {\text{SU}_{q}}(2) $” ''J. Math. Phys.'', '''34''' (1993) pp. 1606–1613.</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top"> C.S. Chu, P.M. Ho, B. Zumino, “The quantum $ 2 $-sphere as a complex manifold” ''Z. Phys. C'', '''70''' (1996) pp. 339–344.</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top"> P. Podles, “Differential calculus on quantum spheres” ''Lett. Math. Phys.'', '''18''' (1989) pp. 107–119.</td></tr>
<tr><td valign="top">[a6]</td> <td valign="top"> M. Noumi, K. Mimachi, “Quantum $ 2 $-spheres and big $ q $-Jacobi polynomials” ''Comm. Math. Phys.'', '''128''' (1990) pp. 521–531.</td></tr>
<tr><td valign="top">[a7]</td> <td valign="top"> S.L. Woronowicz, “Twisted $ \text{SU}(2) $ group. An example of a non-commutative differential calculus” ''Publ. RIMS Univ. Kyoto'', '''23''' (1987) pp. 117–181.</td></tr>
</table>