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Braided Tensor Categories of Admissible Modules for Affine Lie Algebras

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Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is \({\widehat{\mathfrak{sl}}_2}\), we prove the rigidity and modularity conjectures.

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Acknowledgements

T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460). J. Y. is supported in part by an AMS-Simons travel Grant. J. Y. also wants to thank Zongzhu Lin for useful conversations.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Thomas Creutzig

  2. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ, 08854-8019, USA

    Yi-Zhi Huang

  3. Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT, 06511, USA

    Jinwei Yang

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  1. Thomas Creutzig
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  2. Yi-Zhi Huang
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  3. Jinwei Yang
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Correspondence to Jinwei Yang.

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Communicated by Y. Kawahigashi

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Creutzig, T., Huang, YZ. & Yang, J. Braided Tensor Categories of Admissible Modules for Affine Lie Algebras. Commun. Math. Phys. 362, 827–854 (2018). https://doi.org/10.1007/s00220-018-3217-6

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