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Let $I$ be an ideal of a Lie algebra $L$. By the PBW theorem, we can consider $U(I)$ as a subring of $U(L)$. Is it true that $U(I)$ is an ideal of the ring $U(L)$? If not, what kind of object it is?

Luka
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  • Seems like a homework question – pancini Oct 20 '23 at 03:33
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    No, $U(I)$ always contains the identity. – Qiaochu Yuan Oct 20 '23 at 03:57
  • @pancini This is not a hw problem... I was thinking about an easier proof of the invariance lemma and thought the proof will be easier if $U(I)$ is an ideal. – Luka Oct 20 '23 at 04:21
  • @QiaochuYuan Then does this unital subring $U(I)$ have some special property distinguished from other subrings? – Luka Oct 20 '23 at 04:24
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    Because $I$ is a Lie subalgebra, $U(I)$ is a Hopf subalgebra. And the fact that $I$ is an ideal means in addition that $U(I)$ is invariant under the commutator action of $L$ on $U(L)$. – Qiaochu Yuan Oct 20 '23 at 04:29

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