Quantum sphere
A -algebra
generated by two elements
and
satisfying [a1]
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Here, is a deformation parameter and
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
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where stands for the quantum group
[a7] (cf. also Quantum groups) considered as a deformation of the Poisson algebra
. The one-parameter family of quantum spheres is in correspondence with the family of
-covariant Poisson structures on
, which is known to be one-parametric too ([a2], Appendix). The deformation of the Poisson structure
can be introduced in a precisely defined manner [a2]. Also, the structure of representations of
is in correspondence with the structure of symplectic leaves on
[a1], [a2]. For
, the symplectic leaves are two open discs and the points of a circle separating them. For
, one disc leaf is attached to one one-point leaf and, in fact, this is the Bruhat decomposition of the Poisson homogeneous space
. For
,
is a symplectic manifold.
The symplectic spheres , with
, can be realized as orbits of the dressing transformation of
acting on its dual Poisson Lie group. An equivalent realization is given by the right
-action on the manifold
of
unimodular positive matrices, which is just the unitary transformation
. There exists a quantum analogue as a right co-action
, which is defined formally in the same way as in the Poisson case. If
,
, one can construct, using this structure, the
-dimensional irreducible representation of the deformed universal enveloping algebra
[a3]. Moreover, if
, then the
-algebra
is non-trivial only for
[a1].
A few other concepts have been developed for quantum spheres, including a description in terms of a local holomorphic coordinate and its adjoint
[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
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and expressed in terms of big -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 ![]() ![]() |
[a3] | P. Šťovíček, "Quantum line bundles on ![]() ![]() |
[a4] | C.S. Chu, P.M. Ho, B. Zumino, "The quantum ![]() |
[a5] | P. Podles, "Differential calculus on quantum spheres" Lett. Math. Phys. , 18 (1989) pp. 107–119 |
[a6] | M. Noumi, K. Mimachi, "Quantum ![]() ![]() |
[a7] | S.L. Woronowicz, "Twisted ![]() |
Quantum sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_sphere&oldid=36737