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. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.