Abstract
Let [Formula: see text] be a finite-dimensional simple Lie algebra over [Formula: see text]. A [Formula: see text]-module is said to be weight if it is a weight [Formula: see text]-module. We give a complete classification of simple weight modules for [Formula: see text] which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical [Formula: see text]-representation theory, we show that there exists a class of [Formula: see text] irreducible modules which have uniformly two-dimensional weight spaces.
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