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Weight Representations of Admissible Affine Vertex Algebras

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Abstract

For an admissible affine vertex algebra \({V_k{(\mathfrak{g})}}\) of type A, we describe a new family of relaxed highest weight representations of \({V_k{(\mathfrak{g})}}\). They are simple quotients of representations of the affine Kac–Moody algebra \({\widehat{\mathfrak{g}}}\) induced from the following \({\mathfrak{g}}\)-modules: (1) generic Gelfand–Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from \({\mathfrak{sl}_2}\); (2) all Gelfand–Tsetlin modules in the principal nilpotent orbit that are induced from \({\mathfrak{sl}_3}\); (3) all simple Gelfand–Tsetlin modules over \({\mathfrak{sl}_3}\). This in particular gives the classification of all simple positive energy weight representations of \({V_k{(\mathfrak{g})}}\) with finite dimensional weight spaces for \({\mathfrak{g}=\mathfrak{sl}_3}\).

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Tomoyuki Arakawa

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo-SP, Brazil

    Vyacheslav Futorny

  3. Universidade Federal do ABC, Santo André-SP, Brazil

    Luis Enrique Ramirez

Authors
  1. Tomoyuki Arakawa
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  2. Vyacheslav Futorny
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  3. Luis Enrique Ramirez
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Correspondence to Tomoyuki Arakawa.

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Communicated by Y. Kawahigashi

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Arakawa, T., Futorny, V. & Ramirez, L.E. Weight Representations of Admissible Affine Vertex Algebras. Commun. Math. Phys. 353, 1151–1178 (2017). https://doi.org/10.1007/s00220-017-2872-3

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