Abstract
The aim of this note is to show that the affine Lie algebraA (1)1 has a natural family πμ, υ,v of Fock representations on the spaceC[x i,y j;i ∈ ℤ andj ∈ ℕ], parametrized by (μ,v) ∈C 2. By corresponding the highest weightΛ μ, υ of πμ, υ to each (μ,ν), the parameter spaceC 2 forms a double cover of the weight spaceCΛ0⊕C −1 with singularities at linear forms of level −2; this number is (−1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA (1)1 -modules for generic (μ,v).
Similar content being viewed by others
References
Jakobsen, H.P., Kac, V.G.: A new class of unitarizable highest weight representations of infinite dimensional Lie algebras. In: Non-linear equations in classical and quantum field theory. Sanchez (ed.). Lecture Notes in Physics, Vol. 226, pp. 1–20. Berlin, Heidelberg, New York: Springer 1985
Jantzen, J.C.: Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren. Math. Ann.226, 53–65 (1977)
Jantzen, J.C.: Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, Vol. 750. Berlin, Heidelberg, New York: Springer 1979
Kac, V.G.: Infinite dimensional Lie algebras. An introduction. Prog. Math., Boston, Vol.44. Boston: Birkhäuser 1983
Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite dimensional Lie algebras. Adv. Math.34, 97–108 (1979)
Shapovalov, N.N.: On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra. Funct. Anal. Appl.6, 307–312 (1972)
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Wakimoto, M. Fock representations of the affine Lie algebraA (1)1 . Commun.Math. Phys. 104, 605–609 (1986). https://doi.org/10.1007/BF01211068
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01211068