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Let $\mathfrak{g}$ be a Lie algebra. The universal enveloping algebra $U\mathfrak{g}$ is then an infinite-dimensional associative algebra which can be endowed with the structure of a Lie algebra. Is there an infinite-dimensional Lie group whose Lie algebra is $U\mathfrak{g}$? If so, what does this group look like?

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    $\begingroup$ Could you specify what you call an "infinite-dimensional Lie group"? The usual definition starts from a Banach manifold, so that the Lie algebra is a complete normable algebra — which $U(\mathfrak{g}$ is not. $\endgroup$
    – abx
    Feb 10, 2022 at 6:45
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    $\begingroup$ The enveloping algebra gives you the invariant differential operators on the Lie group. $\endgroup$
    – Julian Seipel
    Feb 10, 2022 at 8:36

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