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Proceedings of the American Mathematical Society

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Weight spaces of Lie algebra modules


Author: Martha K. Smith
Journal: Proc. Amer. Math. Soc. 95 (1985), 524-526
MSC: Primary 17B10
MathSciNet review: 810156
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Abstract: Let $ V$ be a finite-dimensional module for the finite-dimensional Lie algebra $ L$ over a field of characteristic zero. If $ {V^\lambda } = \{ v \in V\vert\;{\text{all }}x \in L,{[x - \lambda (x)]^i}v = 0\;{\text{for some }}\} $ is nonzero, then $ \lambda \in {L^*}$ and is a character of $ L$. Moreover, the corresponding eigenspace $ \{ v \in V\vert{\text{all }}x \in L,xv = \lambda (x)v\} $ is nonzero and $ {V^\lambda }$ is an $ L$ submodule of $ V$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0810156-6
Keywords: Finite-dimensional Lie algebra, Lie algebra module, weight space
Article copyright: © Copyright 1985 American Mathematical Society