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[Merged by Bors] - feat(algebra/universal_enveloping_algebra): construction of universal enveloping algebra and its universal property #4041
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If this is merged, it might be worth a follow-up PR grouping some of the Lie algebra theory into its own folder. E.g., something along the lines of:
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Could we have an @[ext] lemma, stating that to show that two algebra maps out of the enveloping algebra are equal, it suffices to show the precompositions with the inclusion are equal? (Perhaps tensor_algebra needs this as well!) |
Agreed! I'm afraid I'm heading away on holiday and will have phone-only internet access for the next week. I tried to squeeze your worthy suggestions in this morning but have run out of time. I got this far: @[ext] lemma hom_ext {g₁ g₂ : universal_enveloping_algebra R L →ₐ[R] A}
(h : ∀ x, g₁ (ι R L x) = g₂ (ι R L x)) : g₁ = g₂ :=
begin
let f₁ := (lie_algebra.of_associative_algebra_hom g₁).comp (ι R L),
let f₂ := (lie_algebra.of_associative_algebra_hom g₂).comp (ι R L),
have h' : f₁ = f₂, by { sorry, }, -- Should be: ext x, exact (h x),
have h₁ : g₁ = lift R L A f₁, { apply lift_unique, refl, },
have h₂ : g₂ = lift R L A f₂, { apply lift_unique, refl, },
rw [h₁, h₂, h'],
end To my embarrassment, extensionality for Lie algebra morphisms is missing! Assuming nobody else finishes this, I'll take this up in a week when I get back. |
Because |
Hopefully extensionality for Lie algebra morphisms is just a matter of putting |
In the meantime, looking again at We should have a look at #4079, and rewrite this PR based on that model. I'm happy to help/do it at some point, but I'll leave this with you for now. |
I’m writing this on a phone while away but I will gratefully refactor along the lines you suggest when I get back on Sunday. Perhaps by then the points raised by @adamtopaz on Zulip will have been settled. |
The I think the same sort of proof, using |
As noted above, we want to refactor this to use
I'm happy to do any of:
though option 1 is probably my least favourite given that being the fourth link in a chain of dependent PRs might mean quite a bit of rebasing etc. |
If you don't object to waiting, I think option 2. is fine. In the meantime, if you want to add some review comments on that trio, and/discuss on zulip, that might move things along. :-) |
@ocfnash, that stack of PRs has now hit master, so I've un-marked this PR as "blocked". Let me know if you want any help with it! |
Excellent thanks. I feel a bit guilty at not helping with the review. I'll have some time this afternoon and will update this PR. I think I shouldn't need help but I will keep the offer in mind :-) |
… enveloping algebra and its universal property
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I've decided I like this approach. I'd also like to dig into the typeclass inference issues that I was whining about on Zulip. |
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Looks great. Just some minor nitpicks on the comments. bors d+ |
✌️ ocfnash can now approve this pull request. To approve and merge a pull request, simply reply with |
Can we wait for #4252 before merging? |
Absolutely. I'll wait for that to land, merge in what it brings to this branch, drop my then-redundant |
…_comm_group and semiring_to_ring (#4252)
…mmit) Also minor proof golfing in `hom_ext`
bors merge |
… enveloping algebra and its universal property (#4041) ## Main definitions * `universal_enveloping_algebra` * `universal_enveloping_algebra.algebra` * `universal_enveloping_algebra.lift` * `universal_enveloping_algebra.ι_comp_lift` * `universal_enveloping_algebra.lift_unique` * `universal_enveloping_algebra.hom_ext` Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Pull request successfully merged into master. Build succeeded: |
Main definitions
universal_enveloping_algebra
universal_enveloping_algebra.algebra
universal_enveloping_algebra.lift
universal_enveloping_algebra.ι_comp_lift
universal_enveloping_algebra.lift_unique
universal_enveloping_algebra.hom_ext