Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 059, 11 pages      arXiv:0806.1976
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras

Dmitry Fuchs and Constance Wilmarth
Department of Mathematics, University of California, One Shields Ave., Davis CA 95616, USA

Received June 29, 2008, in final form August 24, 2008; Published online August 27, 2008

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of sl2 (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103–113]. In the simpler case of A11 the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154–156].

Key words: Kac-Moody algebras; Verma modules; singular vectors.

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  1. Bernstein I.N., Gel'fand I.M., Gel'fand S.I., Structure of representations generated by highest weight vectors, Funktsional. Anal. i Prilozhen. 5 (1971), no. 1, 1-9 (English transl.: Funct. Anal. Appl. 5 (1971), no. 1, 1-8).
  2. Feigin B.L., Fuchs D.B., Verma modules over the Virasoro algebra, in Topology (Leningrad, 1982), Lecture Notes in Math., Vol. 1060, Springer, Berlin, 1984, 230-245.
  3. Fuchs D., Two projections of singular vectors of Verma modules over the affine Lie algebra A11, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 81-83 (English transl.: Funct. Anal. Appl. 23 (1989), no. 2, 154-156).
  4. Kac V., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  5. Kac V., Kazhdan D., Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), 97-108.
  6. Malikov F.G., Feigin B.L., Fuchs D.B., Singular vectors in Verma modules over Kac-Moody algebras, Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 25-37 (English transl.: Funct. Anal. Appl. 20 (1986), no. 2, 103-113).

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