Communications in Mathematical Physics

, Volume 104, Issue 4, pp 605–609 | Cite as

Fock representations of the affine Lie algebraA 1 (1)

  • Minoru Wakimoto


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[xi,yj;i ∈ ℤ andj ∈ ℕ], parametrized by (μ,v) ∈C2. By corresponding the highest weightΛμ, υ of πμ, υ to each (μ,ν), the parameter spaceC2 forms a double cover of the weight spaceCΛ0C1 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).


Neural Network Statistical Physic Complex System Nonlinear Dynamics Linear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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 1985Google Scholar
  2. 2.
    Jantzen, J.C.: Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren. Math. Ann.226, 53–65 (1977)Google Scholar
  3. 3.
    Jantzen, J.C.: Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, Vol. 750. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  4. 4.
    Kac, V.G.: Infinite dimensional Lie algebras. An introduction. Prog. Math., Boston, Vol.44. Boston: Birkhäuser 1983Google Scholar
  5. 5.
    Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite dimensional Lie algebras. Adv. Math.34, 97–108 (1979)Google Scholar
  6. 6.
    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)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Minoru Wakimoto
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceHiroshima UniversityHiroshimaJapan

Personalised recommendations