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Representation theory of \(\mathcal{W}\)-algebras

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Abstract

We study the representation theory of the \(\mathcal{W}\)-algebra \(\mathcal{W}_k(\bar{\mathfrak{g}})\) associated with a simple Lie algebra \(\bar{\mathfrak{g}}\) at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of \(\mathcal{W}_k(\bar{\mathfrak{g}})\) is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra \(\mathfrak{g}\) of \(\bar{\mathfrak{g}}\). As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of \(\mathcal{W}\)-algebras.

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  1. Department of Mathematics, Nara Women’s University, Kitauoyahigashi-machi, Nara, 630-8506, Japan

    Tomoyuki Arakawa

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  1. Tomoyuki Arakawa
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Mathematics Subject Classification (1991)

17B68, 81R10

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Arakawa, T. Representation theory of \(\mathcal{W}\)-algebras. Invent. math. 169, 219–320 (2007). https://doi.org/10.1007/s00222-007-0046-1

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