## On the structure of weight modules

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- by Ivan Dimitrov, Olivier Mathieu and Ivan Penkov PDF
- Trans. Amer. Math. Soc.
**352**(2000), 2857-2869 Request permission

Erratum: Trans. Amer. Math. Soc.

**356**(2004), 3403-3404.

## Abstract:

Given any simple Lie superalgebra ${\mathfrak {g}}$, we investigate the structure of an arbitrary simple weight ${\mathfrak {g}}$-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra**W**$(n)$. Most of them are simply Levi subalgebras of ${\mathfrak {g}}_{0}$, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.

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## Additional Information

**Ivan Dimitrov**- Affiliation: Department of Mathematics, University of California at Riverside, Riverside, California 92521
- Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095-1555
- Email: dimitrov@math.ucla.edu
**Olivier Mathieu**- Affiliation: Université Louis Pasteur, IRMA, 7 rue René Descartes, 67000 Strasbourg, France
- Email: mathieu@math.u-strasbg.fr
**Ivan Penkov**- Affiliation: Department of Mathematics, University of California at Riverside, Riverside, California 92521
- Email: penkov@math.ucr.edu
- Received by editor(s): October 8, 1997
- Published electronically: February 28, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 2857-2869 - MSC (2000): Primary 17B10
- DOI: https://doi.org/10.1090/S0002-9947-00-02390-4
- MathSciNet review: 1624174