Tensor Structure on the Kazhdan-Lusztig Category for Affine gl(1|1)

Creutzig T, Mcrae R, Yang J (2022)

Publication Type: Journal article

Publication year: 2022

Journal

International Mathematics Research Notices Oxford University Press (OUP): Policy H - Oxford Open Option A

Book Volume: 2022

Pages Range: 12462-12515

Journal Issue: 16

DOI: 10.1093/imrn/rnab080

Abstract

We show that the Kazhdan-Lusztig category KLk of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra {gl}(1|1) 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)}-module in KLk has a projective cover in Ok{fin}, and we determine all fusion rules involving simple and projective objects in Ok{fin}. Then using Knizhnik-Zamolodchikov equations, we prove that KLk and Okfin are rigid. As an application of the tensor supercategory structure on Ok{fin}, we study certain module categories for the affine Lie superalgebra sl(2|1) at levels 1 and -1{2}. In particular, we obtain a tensor category of sl(2|1)-modules at level -1{2} that includes relaxed highest-weight modules and their images under spectral flow.

Involved external institutions

University of Alberta CA Canada (CA) Tsinghua University CN China (CN)

How to cite

APA:

Creutzig, T., Mcrae, R., & Yang, J. (2022). Tensor Structure on the Kazhdan-Lusztig Category for Affine gl(1|1). International Mathematics Research Notices, 2022(16), 12462-12515. https://doi.org/10.1093/imrn/rnab080

MLA:

Creutzig, Thomas, Robert Mcrae, and Jinwei Yang. "Tensor Structure on the Kazhdan-Lusztig Category for Affine gl(1|1)." International Mathematics Research Notices 2022.16 (2022): 12462-12515.

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