Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}'$ its subalgebra. Then the universal enveloping algebra $U(\mathfrak{g}')$ can be canonically embedded into $U(\mathfrak{g})$, that of $\mathfrak{g}$.

Now I'm interested in the reverse direction. Given a subalgebra $Y$ of $U(\mathfrak{g})$, under what conditions on $Y$, $Y$ is the universal enveloping algebra $U(\mathfrak{g}')$ of some Lie subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$?

By considering the center of $U(\mathfrak{g})$, not all subalgebras of $U(\mathfrak{g})$ arise as universal enveloping algebras of Lie subalgebras.

I searched the internet but couldn't find anything related to this question. Much appreciated for any answer or reference.

  • 1
    $\begingroup$ Yoiks, I'd think very few. Did you have in mind any further condition? Stability under some involution, for example? $\endgroup$ – paul garrett Dec 13 '16 at 1:19
  • $\begingroup$ @paulgarrett, thanks for your comment. I don't know too much about universal enveloping algebra and have no candidate conditions in my mind. $\endgroup$ – user1832 Dec 13 '16 at 1:25
  • $\begingroup$ Ok, well, if a subgroup or subalgebra is defined as the commutator of another subgroup or subalgebra, then there is the fairly-obvious condition on the corresponding subalgebra of the universal enveloping algebra. Being the fixed subalgebra or subgroup under an involution is much less definitive, so far as I can see. Do you contemplate a commutator condition? $\endgroup$ – paul garrett Dec 13 '16 at 1:28
  • $\begingroup$ @paulgarrett, so you mean if a subalgebra g_1 is the commutator of another g_2, and Y is the commutator of the universal enveloping subalgebra of g_2, then Y is that of g_1? $\endgroup$ – user1832 Dec 13 '16 at 1:48
  • $\begingroup$ IFF it is generated by a subset of $g$. In reality any useful criterion should take into account what you now about your subalgebra, how it is defined, etc... $\endgroup$ – Bugs Bunny Dec 13 '16 at 13:48

In characteristic $0$, if $Y$ is a Hopf subalgebra of $U(\mathfrak{g})$, then $Y = U(\mathfrak{g}')$ for some Lie subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$ (and conversely, every such subalgebra is a Hopf subalgebra). This is a consequence of Proposition 6.13 and Theorem 5.18 of John W. Milnor and John C. Moore's 1965 paper "On the structure of Hopf algebras."

If you are working over a field of positive characteristic, then the result is also true if you replace the universal enveloping algebra by the restricted enveloping algebra (see Theorem 6.11 of loc. cit.).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.