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.