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Tensor structure on the Kazhdan-Lusztig category for affine gl(1|1)

Thomas Creutzig University of Alberta Robert McRae Tsinghua University Jinwei Yang University of Alberta

Category Theory Mathematical Physics Quantum Algebra Representation Theory mathscidoc:2204.04007

We show that the Kazhdan-Lusztig category KL_k of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1)ˆ has vertex algebraic braided tensor supercategory structure, and that its full subcategory O_k^fin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)ˆ-module in KL_k has a projective cover in O_k^fin, and we determine all fusion rules involving simple and projective objects in O_k^fin. Then using Knizhnik-Zamolodchikov equations, we prove that KL_k and O_k^fin are rigid. As an application of the tensor supercategory structure on O_k^fin, we study certain module categories for the affine Lie superalgebra sl(2|1)ˆ at levels 1 and −1/2. In particular, we obtain a tensor category of sl(2|1)ˆ-modules at level −1/2 that includes relaxed highest-weight modules and their images under spectral flow.
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  • To appear in International Mathematics Research Notices
@inproceedings{thomastensor,
  title={Tensor structure on the Kazhdan-Lusztig category for affine gl(1|1)},
  author={Thomas Creutzig, Robert McRae, and Jinwei Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415173238278193055},
}
Thomas Creutzig, Robert McRae, and Jinwei Yang. Tensor structure on the Kazhdan-Lusztig category for affine gl(1|1). http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415173238278193055.
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