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Quantum Lie superalgebras and q-oscillators. (English) Zbl 0764.17023
Summary: Quantum deformation of the simple Lie superalgebras is formulated applying both the method of the Cartan matrix and the \(R\)-matrix approach. Using the quantum analogue of Bose and Fermi oscillators, the realization of quantum \(sl_ q(n| m)\) generators is given. The \(q\)- oscillators are obtained from the quantum algebra itself by the contraction method. Multimode representations in terms of \(q\)-oscillators require nontrivial couplings between the different modes. Possible applications are outlined.

17B81 Applications of Lie (super)algebras to physics, etc.
81T60 Supersymmetric field theories in quantum mechanics
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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[1] Faddeev, L.D.; Kulish, P.P.; Sklyanin, E.K., LES houches lectures 1982, (), 61
[2] Kulish, P.P.; Reshetikhin, N.Yu.; Kulish, P.P.; Reshetikhin, N.Yu.; Sklyanin, E.K.; Sklyanin, E.K., Zap. nauch. semin. LOMI, J. soviet math., Funk. anal. appl., Funk. anal. appl., 17, 34, (1983), [in Russian]
[3] Drinfeld, V.G., (), 798
[4] Jimbo, M., Lett. math. phys., 11, 247, (1986)
[5] Faddeev, L.D.; Reshetikhin, N.Yu.; Takhtajan, L.A.; Faddeev, L.D.; Reshetikhin, N.Yu.; Takhtajan, L.A., Quantization of Lie groups and Lie algebras, (), 129, (1987), preprint
[6] Faddeev, L.D.; Reshetikhin, N.Yu.; Takhtajan, L.A., Algebra i analys, 1, 178, (1989), [in Russian]
[7] Pasquier, V.; Saleur, H., Spht/88-187, (1988), Saclay preprint
[8] Connes, A., Publ. math. IHES, 62, 257, (1985)
[9] Witten, E.; Witten, E., Nucl. phys. B, Iassns-hep-89/32, 268, 253, (1989), Princeton preprint
[10] Maillet, J.M.; Nijhoff, F., Cern th.5449/89, (1989), CERN preprint
[11] Alvarez-Gaumé, L.; Gomez, C.; Sierra, G.; Alvarez-Gaumé, L.; Gomez, C.; Sierra, G.; Moore, G.; Reshetikhin, N.Yu., Phys. lett. B, Cern th.5369/89, Iassns-hep-89/18, 220, 142, (1989), Princeton preprint
[12] Reshetikhin, N.Yu.; Akutsu, Y.; Wadati, M.; Akutsu, Y.; Wadati, M., Lomi e-4-87, e-17-87, J. phys. soc. jpn., Phys. rep., 180, 247, (1989), preprints
[13] Kulish, P.P.; Kulish, P.P.; Reshetikhin, N.Yu., Rims-615, Lett. math. phys., 18, 143, (1989), Kyoto preprint
[14] Kulish, P.P., Proc. alushta school on quantum field theory, ()
[15] Macfarlane, A.J., J. phys. A, 22, 4581, (1989)
[16] Biedenharn, L.C., J. phys. A, 22, L873, (1989) · Zbl 0708.17015
[17] T. Hayashi, Nagoya University preprint (1989).
[18] Kač, V.; Frappat, L.; Sciarrino, A.; Sorba, P., (), Commun. math. phys., 121, 457, (1989)
[19] Rosso, M.; Burroughs, N., Commun. math. phys., Damtp/r-89/4, 124, 307, (1989), Cambridge preprint
[20] Abe, E., Hopf algebras, (1980), Cambridge U.P Cambridge · Zbl 0476.16008
[21] Leites, D., Sov. math. usp., 35, 3, (1980)
[22] Bazhanov, V.; Shadrinov, A., Teor. mat. fiz., 73, 402, (1987)
[23] de Crombrugghe, M.; Rittenberg, V., Ann. phys. (NY), 151, 99, (1983)
[24] Bernard, D.; Le Clair, A., Pupt-1123, (1989), princeton preprint
[25] Reshetikhin, N.; Semenov-Tian-Shansky, M., Msri03808-89, (1989), Berkeley preprint
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