Abstract
We give an explicit description of all inequivalent Heisenberg subalgebras of the affine Lie algebra\(g\hat l_n (\mathbb{C})\) and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra. The construction uses multicomponent fermionic fields and yields a correspondence between bosons (elements of the Heisenberg subalgebra) and fermions.
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Lepowsky, J., Wilson, R. L.: Commun. Math. Phys.62, 43 (1978)
Kac, V. G., Kazhdan, D. A., Lepowsky, J., Wilson, R. L.: Adv. Math.42, 83 (1981)
Frenkel, I. B., Kac, V. G.: Invent. Math.62, 23 (1980)
Segal, G.: Commun. Math. Phys.80, 301 (1981)
Kac, V. G., Peterson, D. H.: 112 constructions of the basic representation of the loop group ofE 8 In: Proceedings of Symposium on Anomalies, Geometry and Topology. Singapore: World Scientific 1985
Lepowsky, J.: Proc. Natl. Acad. Sci. USA82, 8295 (1985)
Kac, V. g.: Infinite dimensional Lie algebras. Boston: Birkhaüser 1983; second edition: Cambridge: Cambridge University Press 1985
Dixon, L., Harvey, J. A., Vafa, C., Witten, E.: Nucl. Phys.B261, 678 (1985) and Nucl. Phys.B274, 285 (1986)
Peterson, D. H., Kac, V. G.: Proc. Natl. Acad. Sci. USA80, 1778 (1983)
Bergvelt, M. J., ten Kroode, A. P. E.: J. Math. Phys.29, 1308 (1988)
Pressley, A. N., Segal, G.: Loop groups, Oxford: Oxford University Press 1986
Fons ten Kroode: Affine Lie algebras and integrable systems. Thesis, University of Amsterdam. 1988
Kac, V. G., Wakimoto, M.: Exceptional hierarchies of soliton equations. In: Proceedings of symposia in pure mathematics49, (1989)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Publications RIMS18, 1077 (1982)
Kac, V. G., Peterson, D. H.: Lectures on the infinite wedge representation and the MKP-hierarchy, In: Proceedings Summer School on Completely Integrable Systems, Montreal 1985 (Université de Montréal, 1986)
Kostant, B.: Am. J. Math.81, 973 (1959)
Carter, R. W.: Simple groups of Lie type, New York: Wiley 1972
Helgason, S.: Differential geometry, Lie groups and symmetric spaces. New York; Academic Press 1978
Kac, V. G., Raina, A. K.: Bombay lectures on highest weight representations of infinite dimensional Lie algebras. Singapore: World Scientific 1987
Kac, V. G., Wakimoto, M.: Adv. Math.70, 156 (1988)
Goddard, P., Olive, D.: Int. J. Mod. Phys.A1, 303 (1986)
Lepowsky, J., Wilson, R. L.: Invent. Math.77, 199 (1984)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: J. Phys. Soc. Jpn50, 3806 (1981)
Dodd, R. K.: J. Math. Phys.31 (3), 533 (1990)
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Communicated by N. Yu. Reshetikhin
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ten Kroode, F., van de Leur, J. Bosonic and fermionic realizations of the affine algebra\(g\hat l_n \) . Commun.Math. Phys. 137, 67–107 (1991). https://doi.org/10.1007/BF02099117
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DOI: https://doi.org/10.1007/BF02099117