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Lie Superalgebras and Irreducibility of \(A_1^{(1)}\) –Modules at the Critical Level

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Abstract

We introduce the infinite-dimensional Lie superalgebra \({\mathcal{A}}\) and construct a family of mappings from a certain category of \({\mathcal{A}}\) –modules to the category of \({A_1^{(1)}}\) –modules at the critical level. Using this approach, we prove the irreducibility of a large family of \({A_1^{(1)}}\) –modules at the critical level parameterized by \(\chi(z) \in \mathbb{C}((z))\) . As a consequence, we present a new proof of irreducibility of certain Wakimoto modules. We also give natural realizations of irreducible quotients of relaxed Verma modules and calculate characters of these representations.

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References

  1. Adamović D. (1999). Representations of the N =  2 superconformal vertex algebra. Internat. Math. Res. Notices 2: 61–79

    Article  Google Scholar 

  2. Adamović, D.: Representations of the vertex algebra \({{\mathcal W}_{1 + \infty}}\) with a negative integer central charge. Comm. Algebra 29(7), 3153–3166 (2001)

    Google Scholar 

  3. Adamović, D.: A construction of admissible \({A_1^{(1)}}\) –modules of level \({-\tfrac{4}{3}}\) . J. Pure Appl. Algebra 196, 119–134 (2005)

    Google Scholar 

  4. Adamović, D., Milas, A.: Vertex operator algebras associated to the modular invariant representations for \({A_1^{(1)}}\) . Math. Res. Lett. 2, 563–575 (1995)

    Google Scholar 

  5. Berman S., Dong C. and Tan S. (2002). Representations of a class of lattice type vertex algebras. J. Pure Appl. Algebra 176: 27–47

    MathSciNet  Google Scholar 

  6. Dong C. and Lepowsky J. (1993). Generalized vertex algebras and relative vertex operators. Birkhäuser, Boston

    MATH  Google Scholar 

  7. Dong C. and Mason G. (1997). On quantum Galois theory. Duke Math. J. 86: 305–321

    Article  MathSciNet  Google Scholar 

  8. Eholzer W. and Gaberdiel M. R. (1997). Unitarity of rational N =  2 superconformal theories. Commun. Math. Phys. 186: 61–85

    MathSciNet  Google Scholar 

  9. Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, Vol. 88, Providence, RI: Amer. Math. Soc., 2001

  10. Frenkel E. (2005). Lectures on Wakimoto modules, opers and the center at the critical level. Adv. Math. 195: 297–404

    Article  MathSciNet  Google Scholar 

  11. Feigin, B., Frenkel, E.: Representations of affine Kac–Moody algebras and bosonization. In: Physics and mathematics of strings, Singapore. World Scientific, 1990, pp. 271–316

  12. Feigin B. and Frenkel E. (1990). Affine Kac-Moody algebras and semi-infinite flag manifolds. Commun. Math. Phys. 128: 161–189

    Article  ADS  MathSciNet  Google Scholar 

  13. Frenkel, I. B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104, (1993)

  14. Frenkel, I. B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Math. Vol. 134, New York: Academic Press, 1988

  15. Friedan D., Martinec E. and Shenker S. (1986). Conformal invariance, supersymmetry and string theory. Nucl. Phys. B 271: 93–165

    ADS  MathSciNet  Google Scholar 

  16. Feigin B. L., Semikhatov A. M. and Tipunin I. Y. (1998). Equivalence between chain categories of representations of affine sl(2) and N =  2 superconformal algebras. J. Math. Phys. 39: 3865–390

    Article  ADS  MathSciNet  Google Scholar 

  17. Frenkel I. B. and Zhu Y. (1992). Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66: 123–168

    Article  MathSciNet  Google Scholar 

  18. Kac V. (1990). Infinite dimensional Lie algebras. Cambridge Univ. Press, Cambridge

    MATH  Google Scholar 

  19. Kac, V.: Vertex Algebras for Beginners. University Lecture Series. Volume 10, Second Edition, Providence, RI: Amer. Math. Soc., 1998

  20. Kac V. and Kazhdan D. (1979). Structure of representations with highest weight of infinite dimensional Lie algebras. Adv. Math. 34: 97–108

    Article  MathSciNet  Google Scholar 

  21. Kazama Y. and Suzuki H. (1989). New N = 2 superconformal field theories and superstring compactifications. Nucl. Phys. B 321: 232–268

    Article  ADS  MathSciNet  Google Scholar 

  22. Li H. (1996). Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109: 143–195

    Article  MathSciNet  Google Scholar 

  23. Li H. (1997). The phyisical superselection principle in vertex operator algebra theory. J. Alg. 196: 436–457

    Article  Google Scholar 

  24. Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations. Progress in Math. 227, Boston: Birkhäuser, 2004

  25. Meurman, A., Primc, M.: Annihilating fields of standard modules of \(\tilde{sl}(2,\mathcal{C})\) and combinatorial identities. Mem. Amer. Math. Soc. 652, 1999

  26. Szczesny M. (2002). Wakimoto modules for twisted affine Lie algebras. Math. Res. Lett. 9(4): 433–448

    MathSciNet  Google Scholar 

  27. Wakimoto, M.: Fock representations of affine Lie algebra \({A_1^{(1)}}\) . Commun. Math. Phys. 104, 605–609 (1986)

    Google Scholar 

  28. Zhu, Y.: Vertex operator algebras, elliptic functions and modular forms. Ph. D. thesis, Yale University, 1990; Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Dražen Adamović

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  1. Dražen Adamović
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Correspondence to Dražen Adamović.

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Communicated by Y. Kawahigashi

Partially supported by the MZOS grant 0037125 of the Republic of Croatia

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Adamović, D. Lie Superalgebras and Irreducibility of \(A_1^{(1)}\) –Modules at the Critical Level. Commun. Math. Phys. 270, 141–161 (2007). https://doi.org/10.1007/s00220-006-0153-7

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