[Merged by Bors] - feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras by jano-wol · Pull Request #22061 · leanprover-community/mathlib4

jano-wol · 2025-02-18T21:58:54Z

Define the maximal nilpotent ideal of Lie algebras

In this PR, I define the maximal nilpotent ideal of a Lie algebra.
A closely related concept—the radical of a Lie algebra (the maximal solvable ideal)—is already defined in Algebra/Lie/Solvable.lean.

I prove the analogous results for the maximal nilpotent ideal that have been established for the radical.

This work is part of the framework:
Classification of semisimple Lie algebras

github-actions · 2025-02-18T21:59:51Z

PR summary be78cb9ace

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

Declarations diff

+ LieIdeal.isNilpotent_iff_le_maxNilpotentIdeal
+ center_le_maxNilpotentIdeal
+ instIsNilpotentSup
+ instMaxNilpotentSubmoduleIsNilpotent
+ isNilpotent_iff_le_maxNilpotentSubmodule
+ maxNilpotentIdeal
+ maxNilpotentIdealIsNilpotent
+ maxNilpotentIdeal_eq_top_of_isNilpotent
+ maxNilpotentSubmodule
+ maxNilpotentSubmodule_eq_top_of_isNilpotent
- instIsNilpotentAdd

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

ocfnash

Thanks!

I've given a fairly nitpicky review, mostly to highlight some points of style which I think can only be learned through review but this is very nice work.

We'll be able to merge this very quickly after you've looked at these suggestions.

Mathlib/Algebra/Lie/Nilpotent.lean

Co-authored-by: Oliver Nash <[email protected]>

jano-wol · 2025-02-20T11:45:29Z

I’ve addressed all your points—thank you! I really appreciate your precise review. This kind of feedback helps me deliver good-quality features to the community!

ocfnash

Thanks!

One final suggestion but otherwise LGTM

bors d+

Mathlib/Algebra/Lie/Nilpotent.lean

mathlib-bors · 2025-02-20T11:58:38Z

✌️ jano-wol can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

jano-wol · 2025-02-20T12:21:56Z

bors r+

…algebras (#22061) Define the maximal nilpotent ideal of Lie algebras Co-authored-by: jano-wol <[email protected]>

mathlib-bors · 2025-02-20T13:56:59Z

Pull request successfully merged into master.

Build succeeded:

* origin/master: (823 commits) chore(Computability): fix naming of lemmas about Sum.inl, Sum.inr, Sum.casesOn (#22156) feat: `Fintype Ordering` (#22154) chore: more renamings to fit the naming convention (#22148) feat(CategoryTheory): any monomorphism in a Grothendieck abelian category is a transfinite composition of pushouts of monomorphisms in a small family (#22157) chore: rename `{Continuous., continuous_}sum_map` to `sumMap` (#22155) chore: remove initial space followed by `-/` (#22158) chore: make arg in HeightOneSpectrum.valuation explicit (#22139) feat(Data/List): `List.maximum` is monotone (#22091) chore(Combinatorics/SimpleGraph): extract WalkDecomp from Walk (#21981) feat: if a monoid M acts, then so does (s : S) with [SubmonoidClass S M] (#21123) chore: backport changes to `getElem` lemmas (#22146) feat(CategoryTheory): generating monomorphisms in Grothendieck abelian categories (#22150) feat: rename variables to fit doc (#22143) feat(CategoryTheory): IsDetecting.isIso_iff_of_mono (#22135) feat(CategoryTheory): truncations of transfinite compositions (#22149) chore: simplify a proof (#22134) feat(CategoryTheory): monomorphisms are stable under coproducts in Grothendieck abelian categories (#22133) feat(Order): Set.Ici.isSuccLimit_coe (#22103) chore(Analysis/Convex/Normed): split into smaller files (#22015) feat(Algebra/Algebra/Lie): max nilpotent ideal <= radical (#22140) refactor(LinearIndependent): refactor to use LinearIndepOn (#21886) chore: rename some more `Foo.sum_elim` -> `sumElim` (#22130) chore: rename Injective.sum_elim and friends (#22129) chore: fix naming oversight from #22070 (#22128) chore: fix spelling mistakes (#22136) feat(RingTheory/Ideal/Quotient): define transtition map between ring or module quotient by powers of ideal (#21900) fix: initialize_simps_projections print warning when projection data already exists (#20339) chore: rename ContMDiff.sum_{elim,map} (#22131) feat(CategoryTheory): characterization of injective objects in terms of lifting properties (#22104) chore(Lean,Tactic): un-indent some doc-strings (#22118) feat(Algebra/MvPolynomial): add `comp` versions of rename lemmas (#21259) chose: add deprecation (#22124) feat(CategoryTheory/Abelian/GrothendieckCategory): computing colimits in Subobject (#22123) feat(CategoryTheory/Subobject): hasCardinalLT_of_mono (#22122) feat: add `DenseRange.piMap` (#22114) feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras (#22061) chore(Data/Finset): don't import algebra when defining `Finset.card` (#21866) feat: a function on a discrete space is smooth (#22113) feat: commutative group objects in additive categories (#21521) style(Geometry/Manifold): remove superfluous indentation in doc-strings (#22117) chore: rename PushNeg.lean to Push.lean (#22108) feat: add Is{Open,Closed}Embedding.sum_elim (#22070) feat(RingTheory/Perfectoid): define the untilt map and generalize pretilt (#21563) feature(Topology/Algebra/Module/WeakBilin): Linear map from F into the topological dual of E with the weak topology (#21078) feat(CategoryTheory/Abelian): the exact sequence attached to a pushout square (#22110) chore(ChartedSpace.lean): group constructions together (#22107) feat(CategoryTheory/MorphismProperty): more basic API (#22099) chore(Data/Fintype): split `Fintype/Card.lean` (#21840) chore(SetTheory/Cardinal/Cofinality): make `IsStrongLimit` into a structure (#21971) feat(Combinatorics/SimpleGraph): Add a theorem about cliques in induced subgraphs (#20705) ...

@pechersky

* polynomial-sequences: (558 commits) Correct the TODO to mention IsCancelAdd which does exist. Talk about strict monotonicity instead, thanks @pechersky. Mention the TODO on li generalization to semirings. One more dedent. Same injectivity twiddling for basis. nontiviality to simplify this proof, and some dedenting. Hopefully more correct injectivity lemmas matching the naming convention. Fix the empty docstring. Update Mathlib/Algebra/Polynomial/Sequence.lean Fix the variable naming. Apply suggestions from code review chore(Computability): fix naming of lemmas about Sum.inl, Sum.inr, Sum.casesOn (#22156) feat: `Fintype Ordering` (#22154) chore: more renamings to fit the naming convention (#22148) feat(CategoryTheory): any monomorphism in a Grothendieck abelian category is a transfinite composition of pushouts of monomorphisms in a small family (#22157) chore: rename `{Continuous., continuous_}sum_map` to `sumMap` (#22155) chore: remove initial space followed by `-/` (#22158) chore: make arg in HeightOneSpectrum.valuation explicit (#22139) feat(Data/List): `List.maximum` is monotone (#22091) chore(Combinatorics/SimpleGraph): extract WalkDecomp from Walk (#21981) feat: if a monoid M acts, then so does (s : S) with [SubmonoidClass S M] (#21123) chore: backport changes to `getElem` lemmas (#22146) feat(CategoryTheory): generating monomorphisms in Grothendieck abelian categories (#22150) feat: rename variables to fit doc (#22143) feat(CategoryTheory): IsDetecting.isIso_iff_of_mono (#22135) feat(CategoryTheory): truncations of transfinite compositions (#22149) chore: simplify a proof (#22134) feat(CategoryTheory): monomorphisms are stable under coproducts in Grothendieck abelian categories (#22133) feat(Order): Set.Ici.isSuccLimit_coe (#22103) chore(Analysis/Convex/Normed): split into smaller files (#22015) feat(Algebra/Algebra/Lie): max nilpotent ideal <= radical (#22140) refactor(LinearIndependent): refactor to use LinearIndepOn (#21886) chore: rename some more `Foo.sum_elim` -> `sumElim` (#22130) chore: rename Injective.sum_elim and friends (#22129) chore: fix naming oversight from #22070 (#22128) chore: fix spelling mistakes (#22136) feat(RingTheory/Ideal/Quotient): define transtition map between ring or module quotient by powers of ideal (#21900) fix: initialize_simps_projections print warning when projection data already exists (#20339) chore: rename ContMDiff.sum_{elim,map} (#22131) feat(CategoryTheory): characterization of injective objects in terms of lifting properties (#22104) chore(Lean,Tactic): un-indent some doc-strings (#22118) feat(Algebra/MvPolynomial): add `comp` versions of rename lemmas (#21259) chose: add deprecation (#22124) feat(CategoryTheory/Abelian/GrothendieckCategory): computing colimits in Subobject (#22123) feat(CategoryTheory/Subobject): hasCardinalLT_of_mono (#22122) feat: add `DenseRange.piMap` (#22114) feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras (#22061) chore(Data/Finset): don't import algebra when defining `Finset.card` (#21866) feat: a function on a discrete space is smooth (#22113) feat: commutative group objects in additive categories (#21521) ...

Define maxNilpotentIdeal

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github-actions bot added the new-contributor This PR was made by a contributor with at most 5 merged PRs. Welcome to the community! label Feb 18, 2025

jano-wol changed the title ~~feat(Algebra/Algebra/Lie): Define the maximal nilpotent ideal of LieAlgebras~~ feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of LieAlgebras Feb 18, 2025

jano-wol changed the title ~~feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of LieAlgebras~~ feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras Feb 18, 2025

github-actions bot added the t-algebra Algebra (groups, rings, fields, etc) label Feb 18, 2025

ocfnash reviewed Feb 20, 2025

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jano-wol and others added 9 commits February 20, 2025 11:57

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Co-authored-by: Oliver Nash <[email protected]>

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Co-authored-by: Oliver Nash <[email protected]>

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Co-authored-by: Oliver Nash <[email protected]>

Add -> Sup

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max_nilpotent_ideal -> maxNilpotentIdeal

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feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie …

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…algebras (#22061) Define the maximal nilpotent ideal of Lie algebras Co-authored-by: jano-wol <[email protected]>

mathlib-bors bot changed the title ~~feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras~~ [Merged by Bors] - feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras Feb 20, 2025

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[Merged by Bors] - feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras#22061

[Merged by Bors] - feat(Algebra/Algebra/Lie): define the maximal nilpotent ideal of Lie algebras#22061
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