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Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \mathfrak{g}'$ as Lie algebras.

I am looking for examples such that $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as algebras but $\mathfrak{g}\not\cong \mathfrak{g}'$ as Lie algebras (over an algebraically closed field). Moreover, are there examples such that the categories $U(\mathfrak{g})-\text{Mod}$ and $U(\mathfrak{g}')-\text{Mod}$ are not monoidally equivalent?

I'm not very familiar with the isomorphism problem for enveloping algebras, a quick google search only gave me counterexamples in positive characteristic. I'd be very happy with examples in characteristic zero (infinite dimensions are allowed). I'm more into the monoidal stuff and might figure out myself whether the representation categories are monoidally equivalent.

Edit: I'm asking this because I naturally encountered a quantized version of this problem. Obviously the categories $U(\mathfrak{g})-\text{Mod}$ and $U(\mathfrak{g}')-\text{Mod}$ are Morita equivalent but there is more information here. First of all $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as algebras which clearly is stronger but they are also enveloping algebras of Lie algebras, further restricting possibilities. In the quantized version I'm looking at, I suspect the representation rings of both categories to be the same making the difference in the monoidal structure very subtle. So I'm wondering whether anything on this subject is known in the non-quantized world.

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We have recently settled this problem in the article Lie, associative, and commutative quasi-isomorphism (with R. Campos, D. Petersen, and F. Wierstra, comments are welcome!).

The result in which you are interested is:

Theorem B - classical version: Over a field of characteristic $0$, two Lie algebras (non-dg) are isomorphic if and only if their universal enveloping algebras are isomorphic as associative algebras.

This is a consequence of a more general result (Theorem B) stating that two dg Lie algebras are quasi-isomorphic if and only if their universal enveloping algebras are quasi-isomorphic as associative dg algebras.

In my view (my coauthors might disagree) the spirit of the proof is mostly deformation theoretical, but operad theory play a big supporting role. Its structure is explained in paragraphs 0.22-0.24.

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FWIW, a ten year old article states: "We stress that, in spite of all this, the characteristic zero case of the isomorphism problem remains entirely open."

(https://link.springer.com/article/10.1007/s10468-007-9083-0)

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  • $\begingroup$ What a shame, this probably also explains why I was completely stuck on the quantized problem I encountered. The quantized version shouldn't be too much different as the representation theory of $\mathfrak{g}$ and $U_v(\mathfrak{g})$ are often very similar. Anyway, in the near future we will publish an article on something completely different where all of the sudden this question pops up. Maybe smarter people can use our example to book some progress. $\endgroup$ – Mathematician 42 Feb 6 '18 at 8:59
  • $\begingroup$ Thanks btw, I encountered this paper but completely missed the sentence saying that the characteristic zero case is still open. Thank you for spotting this. $\endgroup$ – Mathematician 42 Feb 6 '18 at 9:12

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