Quantum line bundles on \(S^ 2\) and method of orbits for \(SU_ q(2)\). (English) Zbl 0782.17010
The aim of this paper is to construct representations of the quantum group \(SU_ q(2)\) for \(q\in(0,1)\) using an analog of the orbit method (geometric quantization). First the author studies the relations between functions on \(SU_ q(2)\), the deformed universal enveloping algebra of \({\mathfrak {sl}}(2,\mathbb{C})\) and functions on \(AN_ q\), where \(AN\) denotes the solvable part in the Iwasawa decomposition of \(SL(2,\mathbb{C})\). Next he discusses quantum dressing orbits, which turn out to be quantum two- spheres and constructs the quantum analog of the cover of a two-sphere by two planes. Using this covering he constructs line bundles over the quantum sphere and via an analog of the classical reduction using a polarization he arrives at irreducible representations on spaces of “holomorphic sections” of these bundles. Finally the classical limit of these constructions and possible generalizations are discussed.
Reviewer: A.Cap (Wien)
MSC:
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
46L85 | Noncommutative topology |
46L87 | Noncommutative differential geometry |
16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
quantum dressing transformation; geometric quantization; quantum line bundles; representations; quantum group; orbit method; quantum sphereReferences:
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