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Consider a finite-dimensional Lie algebra over C. Sometimes (e.g. for any semisimple Lie algebra) the center of its universal enveloping algebra is isomorphic to a freely generated polynomial ring.

What can be said in general about the center of the universal enveloping algebra? In which case is it freely generated? What are the degrees of the generators?

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    $\begingroup$ What do you mean by "freely generated"? $\endgroup$
    – YCor
    Commented Nov 16, 2022 at 21:35
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    $\begingroup$ The center of the universal enveloping algebra of ${\mathfrak g}$ is the ring of invariants of its symmetric algebra under the action of ${\mathfrak g}$, the Duflo isomorphism. $\endgroup$
    – F Zaldivar
    Commented Nov 16, 2022 at 23:23
  • $\begingroup$ Thanks for the comment, but this relates one problem to another problem. $\endgroup$
    – Carlo
    Commented Nov 17, 2022 at 12:38
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    $\begingroup$ I do not think there is any hope for any very general statements. For some related results, see sciencedirect.com/science/article/pii/S0021869312002475 and references therein. $\endgroup$
    – Vladimir Dotsenko
    Commented Nov 17, 2022 at 18:13

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