Glasnik Matematicki, Vol. 48, No. 1 (2013), 81-90.

A NOTE ON REPRESENTATIONS OF SOME AFFINE VERTEX ALGEBRAS OF TYPE D

Ozren Perše

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: perse@math.hr

Abstract.   In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to D l(1) of levels n-l+1, for n Z >0. For n=1, we study the representation theory of the quotient vertex operator algebra modulo the ideal generated by that singular vector. In the case l =4, we show that the adjoint module is the unique irreducible ordinary module for simple vertex operator algebra LD4(-2,0). We also show that the maximal ideal in associated universal affine vertex algebra is generated by three singular vectors.

2010 Mathematics Subject Classification.   17B69, 17B67, 81R10.

Key words and phrases.   Vertex operator algebra, affine Kac-Moody algebra, Zhu's algebra.

Full text (PDF) (free access)

DOI: 10.3336/gm.48.1.07

References:

  1. D. Adamović, Some rational vertex algebras, Glas. Mat. Ser. III 29(49) (1994), 25-40.
    MathSciNet    

  2. D. Adamović, A construction of some ideals in affine vertex algebras, Int. J. Math. Math. Sci. 2003 (2003), 971-980.
    MathSciNet     CrossRef

  3. D. Adamović, A construction of admissible A1(1)-modules of level -4/3, J. Pure Appl. Algebra 196 (2005), 119-134.
    MathSciNet     CrossRef

  4. D. Adamović and A. Milas, Vertex operator algebras associated to modular invariant representations for A1(1), Math. Res. Lett. 2 (1995), 563-575.
    MathSciNet     CrossRef

  5. D. Adamović and O. Perše, Some general results on conformal embeddings of affine vertex operator algebras, Algebr. Represent. Theory 16 (2013), 51-64.
    MathSciNet     CrossRef

  6. J. D. Axtell and K.-H. Lee, Vertex operator algebras associated to type G affine Lie algebras, J. Algebra 337 (2011), 195-223.
    MathSciNet     CrossRef

  7. R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 3068-3071.
    MathSciNet     CrossRef

  8. N. Bourbaki, Iliments de mathimatique. (French) Fasc. XXXVIII: Groupes et alg.bres de Lie. Chapitre VII: Sous-alg.bres de Cartan, iliments riguliers. Chapitre VIII: Alg.bres de Lie semi-simples diployies. Actualitis Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975.

  9. C. Dong, H. Li and G. Mason, Vertex operator algebras associated to admissible representations of , Comm. Math. Phys. 184 (1997), 65-93.
    MathSciNet     CrossRef

  10. A. J. Feingold and I. B. Frenkel, Classical affine algebras, Adv. in Math. 56 (1985), 117-172.
    MathSciNet     CrossRef

  11. E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves, American Mathematical Society, Providence, 2001.
    MathSciNet    

  12. I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993.
    MathSciNet    

  13. I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster, Academic Press, Boston, 1988.
    MathSciNet    

  14. I. Frenkel and Y.-C. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
    MathSciNet     CrossRef

  15. V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
    MathSciNet     CrossRef

  16. V. G. Kac, Vertex algebras for beginners, AMS, Providence, 1998.
    MathSciNet    

  17. V. Kac and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), 4956-4960.
    MathSciNet     CrossRef

  18. J. Lepowsky and H. Li, Introduction to vertex operator algebras and their representations, Birkhäuser, Boston, 2004.
    MathSciNet     CrossRef

  19. H.-S. Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), 143-195.
    MathSciNet     CrossRef

  20. A. Meurman and M. Primc, Annihilating fields of standard modules of sl(2, C)~ and combinatorial identities, Mem. Amer. Math. Soc. 137, AMS, Providence RI, 1999.
    MathSciNet    

  21. O. Perše, Vertex operator algebras associated to type B affine Lie algebras on admissible half-integer levels, J. Algebra 307 (2007), 215-248.
    MathSciNet     CrossRef

  22. O. Perše, Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A, Glas. Mat. Ser. III 43(63) (2008), 41-57.
    MathSciNet     CrossRef

  23. Y.-C. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.
    MathSciNet     CrossRef
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