From previous posts, I can see that the ideals of a Lie algebra $L$ are exactly the submodules of $L$ when we make a Lie algebra $L$ into an $L$-module via the adjoint homomorphism. But I am trying to understand if this holds for a general module $V$. Given a Lie algebra $L$ and an $L$-module $V$, can we say that the submodules of $V$ are exactly the ideals of the Lie algebra $L$?
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3The question doesn't make sense: the submodules of $V$ are subsets of $V$, whereas the ideals of $L$ are subsets of $L$. How can they possibly be equal? – José Carlos Santos Jan 01 '23 at 13:31
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I am still new to Lie algebra and Representation Theory. I have been studying Representation of a Lie algebra for my undergraduate project for less than three months. Would that make sense if I say submodules of $L$ instead of $V$? – Askaaren Jan 01 '23 at 13:42
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Yes, it would.${}$ – José Carlos Santos Jan 01 '23 at 13:49
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1See this post. If $V$ is not the adjoint module, we do not have such a relation. – Dietrich Burde Jan 01 '23 at 15:47