Quantum sphere

A $C^{*}$-algebra $\text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right)$ generated by two elements $A$ and $B$ satisfying [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 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 [a7] (cf. also Quantum groups) considered as a deformation of the 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 ([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 [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 $\Bbb{S}^{2}_{c}$ [a1], [a2]. For $c > 0$, 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 $c < 0$, $\Bbb{S}^{2}_{c}$ is a symplectic manifold.

The symplectic spheres $\Bbb{S}^{2}_{c}$, with $c < 0$, can be realized as orbits of the dressing transformation of $\text{SU}(2)$ acting on its dual 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 ${\mathcal{U}_{q}}(\mathfrak{su}(2))$ [a3]. Moreover, if $c < 0$, then the $C^{*}$-algebra $\text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right)$ is non-trivial only for $c = c(n)$ [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 $z^{*}$ [a3], [a4] and a differential and integral calculus [a4], [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 [a6].

References

[a1]	P. Podles, “Quantum spheres” Lett. Math. Phys., 14 (1987) pp. 193–202.
[a2]	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.
[a3]	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.
[a4]	C.S. Chu, P.M. Ho, B. Zumino, “The quantum $2$-sphere as a complex manifold” Z. Phys. C, 70 (1996) pp. 339–344.
[a5]	P. Podles, “Differential calculus on quantum spheres” Lett. Math. Phys., 18 (1989) pp. 107–119.
[a6]	M. Noumi, K. Mimachi, “Quantum $2$-spheres and big $q$-Jacobi polynomials” Comm. Math. Phys., 128 (1990) pp. 521–531.
[a7]	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.