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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Simple supermodules over Lie superalgebras


Authors: Chih-Whi Chen and Volodymyr Mazorchuk
Journal: Trans. Amer. Math. Soc. 374 (2021), 899-921
MSC (2010): Primary 16E30, 17B10
DOI: https://doi.org/10.1090/tran/8303
Published electronically: December 3, 2020
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Abstract: We show that, for many Lie superalgebras admitting a compatible $ \mathbb{Z}$-grading, the Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra. This reduces the classification problem for the former to the one for the latter. Our result applies to all classical Lie superalgebras of type $ I$, in particular, to the general linear Lie superalgebra $ \mathfrak{gl}(m\vert n)$. In the latter case we also show that the rough structure of simple $ \mathfrak{gl}(m\vert n)$-supermodules and also that of Kac supermodules depends only on the annihilator of the $ \mathfrak{gl}(m)\oplus \mathfrak{gl}(n)$-input and hence can be computed using the combinatorics of BGG category $ \mathcal {O}$.


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Additional Information

Chih-Whi Chen
Affiliation: Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden; and Department of Mathematics, National Central University, Chung-Li, Taiwan 32054
Email: cwchen@math.ncu.edu.tw

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden
Email: mazor@math.uu.se

DOI: https://doi.org/10.1090/tran/8303
Received by editor(s): September 27, 2018
Received by editor(s) in revised form: January 14, 2020
Published electronically: December 3, 2020
Additional Notes: The first author was supported by Vergstiftelsen and an MoST grant
The second author was supported by the Swedish Research Council and Göran Gustafsson Stiftelser.
Article copyright: © Copyright 2020 American Mathematical Society