[Merged by Bors] - feat: if a Lie algebra has non-degenerate Killing form then its Cartan subalgebras are Abelian #8430
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The real problem here is that the definition of `DirectSum.IsInternal.collectedBasis` commits definitional abuse by using `DFinsupp.mapRange.linearEquiv` on terms which are of type `DirectSum` rather than `DFinsupp`. One way to avoid this would be to duplicate the API for `DFinsupp.mapRange` in terms of `DirectSum` and use this in `collectedBasis`.
…rMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal'`
…n subalgebras are Abelian Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.
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Can this proof be improved?
ocfnash
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Nov 17, 2023
| @@ -453,6 +453,13 @@ theorem isInternal_submodule_iff_isCompl (A : ι → Submodule R M) {i j : ι} ( | |||
| exact ⟨fun ⟨hd, ht⟩ ↦ ⟨hd, codisjoint_iff.mpr ht⟩, fun ⟨hd, ht⟩ ↦ ⟨hd, ht.eq_top⟩⟩ | |||
| #align direct_sum.is_internal_submodule_iff_is_compl DirectSum.isInternal_submodule_iff_isCompl | |||
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| -- Controversial lemma (EricW not a fan). Let's use to get sorry-free and then reconsider | |||
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This lemma was excised from a previous PR #8369
@eric-wieser raised some good points about how we should prove this lemma here
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Now that this PR compiles, what is the status here?
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I have removed this code comment in the commit 4f9a599
I believe Eric's position was that it might be worth proving this lemma without using DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top so that we would be able to apply it to additive monoids. I feel this is a large detour considering we have no application in mind and this is only a two-line proof. However I recognise his position is reasonable.
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Ok, then I'm going to merge. We can refactor later if needed.
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Nov 20, 2023
| @@ -453,6 +453,13 @@ theorem isInternal_submodule_iff_isCompl (A : ι → Submodule R M) {i j : ι} ( | |||
| exact ⟨fun ⟨hd, ht⟩ ↦ ⟨hd, codisjoint_iff.mpr ht⟩, fun ⟨hd, ht⟩ ↦ ⟨hd, ht.eq_top⟩⟩ | |||
| #align direct_sum.is_internal_submodule_iff_is_compl DirectSum.isInternal_submodule_iff_isCompl | |||
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| -- Controversial lemma (EricW not a fan). Let's use to get sorry-free and then reconsider | |||
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Now that this PR compiles, what is the status here?
This should be (and now is) raised on GitHub instead of in a code comment
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Thanks 🎉 bors merge |
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…n subalgebras are Abelian (#8430) Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.
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…n subalgebras are Abelian (#8430) Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.
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…n subalgebras are Abelian (#8430) Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.
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…n subalgebras are Abelian (#8430) Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.
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Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.