Abstract
We are defining the trigonometric Lie subalgebras in\(\bar X_\infty = \bar A_\infty (\bar B_\infty ,\bar C_\infty ,\bar D_\infty )\) which are the natural generalization of the well known Sin-Lie algebra. The embedding formulas into\(\bar X_\infty \) are introduced. These algebras can be considered as some Lie algebras of quantum tori. An irreducible representation ofA, B series of trigonometric Lie algebras is constructed. Special cases of the trigonometric Lie factor algebras, which can be considered as a quantum (preserving Lie algebra structure) deformation of the Kac-Moody algebras are considered.
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Fairlie, D., Fletcher, P., Zachos, C.: Phys. Lett.218 B, 203 (1989)
Weyl, H.: The theory of groups and quantum mechanics. Dover, 1931
Moyal, J.: Proc. Camb. Phil. Soc.45, 99 (1949)
Arnol'd, V.: Ann. Inst. Fourier16, 319 (1966)
Pimsner, M., Voiculescu, D.: J. Op. Theory4, 201 (1980)
Connes, A.: C.R. Acad. Sci. Paris t.290. Serie A, 599 (1980)
Manin, Yu.: Quantum groups and non-communtative geometry, CRM Université de Montréal, 1988
Hoppe, J.: Phys. Lett. B215, 706 (1988)
Floratos, E.G.: Phys. Lett. B232, 467 (1989)
Golenischeva-Kutuzova, M., Lebedev, D.: Preprint Bonn-HE-90-09; Soviet JETP Lett.52, 1164 (1990)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. Proc. RIMS Sympos. Jimbo, M., Miwa, T. (eds) pp. 39–120. Singapore: World Scientific 1983
Jimbo, M., Miwa, T.: Solitons and infinite-dimensional algebras. Publ. Res. Inst. Math. Sci.19, 943 (1983)
Kac, V.: Infinite-dimensional Lie algebras. Cambridge: Cambridge University Press 1985
Ueno, K., Takasaki, K.: Adv. Stud. Pure Math.4, 1 (1984)
Bogoyavlesky, O.I.: Commun. Math. Phys.51, 201 (1976)
Mikhailov, A.V., Olshanetsky, M.A., Perelomov, A.M.: Commun. Math. Phys.79, 473 (1981)
Lebedev, D., Orlov, A., Pakuliak, S., Zabrodin, A.: Preprint Bonn-HE-91-05; Phys. Lett. A160, 166 (1991).
Degasperis, A., Lebedev, D., Olshanetsky, M., Pakuliak, S., Perelomov, A., Santini, P.: Commun. Math. Phys.141, 133–151 (1991)
Lebedev, D., Pakuliak, S.: Phys. Lett. A160, 173 (1991)
Lepowsky, J., Wilson, R.: Commun. Math. Phys.62, 43 (1978)
Kac, V., Kazhdan, D., Lepowsky, J., Wilson, R.: Adv. Math.42, 83 (1981)
Pope, C.N., Romans, L.J.: Class. Quantum Grav.7, 97 (1990)
Saveliev, M.V., Verchik, A.M.: Phys. Lett. A143, 121 (1990); Commun. Math. Phys.126, 367 (1989)
Konstein, S.E., Vasiliev, M.A.: Nucl. Phys. B312, 402 (1989)
Gerasimov, A., Lebedev, D., Morozov, A.: Preprint ITEP 4-90 (1990); J. Mod. Phys. A6, 977 (1991)
Golenishcheva-Kutuzova, M., Lebedev, D.: Soviet JETP Lett.54, 473 (1991)
Frenkel, I.B., Jing, N.: Proc. Natl. Acad. Sci., USA85,9373 (1988)
Connes, A.: Publ. Math. IHES62, 257–360 (1985)
Connes, A., Rieffel, M.: Contemp. Math.62, 237 (1987)
Hoppe, J., Olshanetsky, M., Theisen, S.: Dynamical systems on quantum tori algebras. Preprint KA-THEP-10/91
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Communicated by N. Yu. Reshetikhin
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Golenishcheva-Kutuzova, M., Lebedev, D. Vertex operator representation of some quantum tori Lie algebras. Commun.Math. Phys. 148, 403–416 (1992). https://doi.org/10.1007/BF02100868
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DOI: https://doi.org/10.1007/BF02100868