Lie algebra modules with finite-dimensional weight spaces. I

Fernando, S. L.

doi:10.1090/S0002-9947-1990-1013330-8

Lie algebra modules with finite-dimensional weight spaces. I
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Trans. Amer. Math. Soc. 322 (1990), 757-781 Request permission

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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