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Lie algebra modules with finite-dimensional weight spaces. I


Author: S. L. Fernando
Journal: Trans. Amer. Math. Soc. 322 (1990), 757-781
MSC: Primary 17B10
DOI: https://doi.org/10.1090/S0002-9947-1990-1013330-8
MathSciNet review: 1013330
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Abstract: Let $ \mathfrak{g}$ denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let $ \mathfrak{h}$ denote a Cartan subalgebra of $ \mathfrak{g}$. In this paper we study finitely generated $ \mathfrak{g}$-modules that decompose into direct sums of finite dimensional $ \mathfrak{h}$-weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if $ \mathfrak{g}$ is a simple Lie algebra that admits a torsion free module, then $ \mathfrak{g}$ is of type $ A$ or $ C$.


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DOI: https://doi.org/10.1090/S0002-9947-1990-1013330-8
Keywords: Reductive Lie algebra, weight module, Gelfand-Kirillov dimension
Article copyright: © Copyright 1990 American Mathematical Society

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