Let $G$ be an algebraic group and $\mathfrak{g}$ its Lie algebra. Are the set of all $G$-modules the same as the set of all $U(\mathfrak{g})$-modules? Thank you very much.
2026-03-29 10:29:34.1774780174
Are the set of all $G$-modules the same as the set of all $U(\mathfrak{g})$-modules?
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The question is ill-posed, as there is no way for them to be "the same".
There is a canonical way (based on differentiating) to construct from a $G$-module a $U(g)$-module. If the group $G$ is simply connected then this is a bijection of isomorphism classes. If not, not.