Simple supermodules over Lie superalgebras
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- by Chih-Whi Chen and Volodymyr Mazorchuk PDF
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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 $\mathfrak {gl}(m|n)$. In the latter case we also show that the rough structure of simple $\mathfrak {gl}(m|n)$-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 $\mathcal {O}$.References
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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
- MR Author ID: 1022852
- Email: cwchen@math.ncu.edu.tw
- Volodymyr Mazorchuk
- Affiliation: Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden
- MR Author ID: 353912
- Email: mazor@math.uu.se
- 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. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 899-921
- MSC (2010): Primary 16E30, 17B10
- DOI: https://doi.org/10.1090/tran/8303
- MathSciNet review: 4196382