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By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?

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    $\begingroup$ Every nilpotent Lie algebra is solvable; is there something more specific that you need about their representations? $\endgroup$ – guest Jan 3 '15 at 20:16

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