On coherent states for the simplest quantum groups. (English) Zbl 0735.17024
Summary: The coherent states for the simplest quantum groups (\(q\)-Heisenberg-Weyl, \(SU_ q(2)\) and the discrete series of representations of \(SU_ q(1,1)\)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and \(q\)- deformation of Berezin’s quantization on \(\mathbb{C}\), a sphere, and the Lobatchevsky plane are discussed.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B15 | Representations of Lie algebras and Lie superalgebras, analytic theory |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
References:
| [1] | KlauderJ. R., A coherent-state primer, in J. R.Klauder and B-S.Skagerstam (eds), Coherent States, World Scientific, Singapore, 1985. |
| [2] | PerelomovA. M., Generalized Coherent States and Their Applications, Springer, New York, 1986. |
| [3] | Drinfel’d, V. G., Quantum groups, in Proc. ICM Berkley (1986) AMS, 1987. |
| [4] | BiedenharnL. C., J. Phys. A 22, 873 (1989). |
| [5] | MacfarlaneA. J., J. Phys. A. 22, 4581 (1989). · Zbl 0722.17009 · doi:10.1088/0305-4470/22/21/020 |
| [6] | BargmannV., Comm. Pure. Appl. Math. 14, 187 (1961). · Zbl 0107.09102 · doi:10.1002/cpa.3160140303 |
| [7] | BerezinF. A., Izv. Akad. Nauk SSSR 36 1134 (1972). |
| [8] | JimboM., Lett. Math. Phys. 10, 63 (1985). · Zbl 0587.17004 · doi:10.1007/BF00704588 |
| [9] | Kirillov, A. N. and Reshetikhin, N. Yu., LOMI preprint, E-9-88, 1988. |
| [10] | RadcliffeJ. M., J. Phys. A. 4, 313 (1971). · doi:10.1088/0305-4470/4/3/009 |
| [11] | AndrewsG. E., The Theory of Partitions, Addison-Wesley., Reading, Mass., 1974. |
| [12] | Vilenkin, N. Ya., Special Functions and Theory of Group Representations, AMS, 1968. · Zbl 0172.18404 |
| [13] | MasudaT., MimachiK., NakagamiY., NuomiM., SaburiY., and UenoK., Unitary representations of quantum group SU q (1, 1) I, II, Lett. Math. Phys. 19, 187, 195 (1990). · Zbl 0704.17008 · doi:10.1007/BF01039312 |
| [14] | BargmannV., Ann. Math. 48, 568 (1947). · Zbl 0045.38801 · doi:10.2307/1969129 |
| [15] | OdzijewiczA., Comm. Math. Phys. 114, 577 (1988). · Zbl 0645.53044 · doi:10.1007/BF01229456 |
| [16] | BerezinF. A., Comm. Math. Phys. 40, 153 (1975). · Zbl 1272.53082 · doi:10.1007/BF01609397 |
| [17] | BerezinF. A., Izv. Akad. Nauk SSSR 38 1116 (1974). |
| [18] | BerezinF. A., Izv. Akad. Nauk SSSR 39, 363 (1975). |
| [19] | TuynmanG. M., J. Math. Phys. 28 573 (1987). · Zbl 0616.58041 · doi:10.1063/1.527642 |
| [20] | Tuynman, G. M., Preprint, Univ. of Amsterdam, Report 87-03, 1987. |
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