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[Merged by Bors] - chore: refactor Lie algebra weight spaces #7210

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[Merged by Bors] - chore: refactor Lie algebra weight spaces #7210

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2d0e3f5
chore: refactor Lie algebra weight spaces
ocfnash Sep 13, 2023
aea6103
Appease `simpNF` linter
ocfnash Sep 16, 2023
e52ee91
Use `local notation3` instead of `local macro_rules` and add doc string.
ocfnash Sep 18, 2023
e5dc93d
Merge branch 'master' into ocfnash/killing_cartan
ocfnash Sep 18, 2023
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23 changes: 5 additions & 18 deletions Mathlib/Algebra/Lie/Nilpotent.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -244,20 +244,13 @@ theorem isNilpotent_toEndomorphism_of_isNilpotent₂ [IsNilpotent R L M] (x y :
rw [LinearMap.pow_apply, LinearMap.zero_apply, ← LieSubmodule.mem_bot (R := R) (L := L), ← hM]
exact iterate_toEndomorphism_mem_lowerCentralSeries₂ R L M x y m k

/-- For a nilpotent Lie module, the weight space of the 0 weight is the whole module.

This result will be used downstream to show that weight spaces are Lie submodules, at which time
it will be possible to state it in the language of weight spaces. -/
theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
⨅ x : L, (toEndomorphism R L M x).maximalGeneralizedEigenspace 0 = ⊤ := by
@[simp] lemma maxGenEigenSpace_toEndomorphism_eq_top [IsNilpotent R L M] (x : L) :
((toEndomorphism R L M x).maximalGeneralizedEigenspace 0) = ⊤ := by
ext m
simp only [Module.End.mem_maximalGeneralizedEigenspace, Submodule.mem_top, sub_zero, iff_true_iff,
zero_smul, Submodule.mem_iInf]
intro x
simp only [Module.End.mem_maximalGeneralizedEigenspace, zero_smul, sub_zero, Submodule.mem_top,
iff_true]
obtain ⟨k, hk⟩ := exists_forall_pow_toEndomorphism_eq_zero R L M
use k; rw [hk]
exact LinearMap.zero_apply m
#align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
exact ⟨k, by simp [hk x]⟩

/-- If the quotient of a Lie module `M` by a Lie submodule on which the Lie algebra acts trivially
is nilpotent then `M` is nilpotent.
Expand Down Expand Up @@ -560,12 +553,6 @@ theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
LieModule.exists_forall_pow_toEndomorphism_eq_zero R L L
#align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra

/-- See also `LieAlgebra.zero_rootSpace_eq_top_of_nilpotent`. -/
theorem LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0 = ⊤ :=
LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
#align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent

-- TODO Generalise the below to Lie modules if / when we define morphisms, equivs of Lie modules
-- covering a Lie algebra morphism of (possibly different) Lie algebras.
variable {R L L'}
Expand Down
19 changes: 18 additions & 1 deletion Mathlib/Algebra/Lie/Submodule.lean
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Expand Up @@ -92,12 +92,16 @@ theorem mem_carrier {x : M} : x ∈ N.carrier ↔ x ∈ (N : Set M) :=
Iff.rfl
#align lie_submodule.mem_carrier LieSubmodule.mem_carrier

@[simp]
theorem mem_mk_iff (S : Set M) (h₁ h₂ h₃ h₄) {x : M} :
x ∈ (⟨⟨⟨⟨S, h₁⟩, h₂⟩, h₃⟩, h₄⟩ : LieSubmodule R L M) ↔ x ∈ S :=
Iff.rfl
#align lie_submodule.mem_mk_iff LieSubmodule.mem_mk_iff

@[simp]
theorem mem_mk_iff' (p : Submodule R M) (h) {x : M} :
x ∈ (⟨p, h⟩ : LieSubmodule R L M) ↔ x ∈ p :=
Iff.rfl

@[simp]
theorem mem_coeSubmodule {x : M} : x ∈ (N : Submodule R M) ↔ x ∈ N :=
Iff.rfl
Expand Down Expand Up @@ -419,6 +423,11 @@ theorem sInf_coe_toSubmodule (S : Set (LieSubmodule R L M)) :
rfl
#align lie_submodule.Inf_coe_to_submodule LieSubmodule.sInf_coe_toSubmodule

@[simp]
theorem iInf_coe_toSubmodule {ι} (p : ι → LieSubmodule R L M) :
(↑(⨅ i, p i) : Submodule R M) = ⨅ i, (p i : Submodule R M) := by
rw [iInf, sInf_coe_toSubmodule]; ext; simp

@[simp]
theorem sInf_coe (S : Set (LieSubmodule R L M)) : (↑(sInf S) : Set M) = ⋂ s ∈ S, (s : Set M) := by
rw [← LieSubmodule.coe_toSubmodule, sInf_coe_toSubmodule, Submodule.sInf_coe]
Expand All @@ -427,6 +436,14 @@ theorem sInf_coe (S : Set (LieSubmodule R L M)) : (↑(sInf S) : Set M) = ⋂ s
and_imp, SetLike.mem_coe, mem_coeSubmodule]
#align lie_submodule.Inf_coe LieSubmodule.sInf_coe

@[simp]
theorem iInf_coe {ι} (p : ι → LieSubmodule R L M) : (↑(⨅ i, p i) : Set M) = ⋂ i, ↑(p i) := by
rw [iInf, sInf_coe]; simp only [Set.mem_range, Set.iInter_exists, Set.iInter_iInter_eq']

@[simp]
theorem mem_iInf {ι} (p : ι → LieSubmodule R L M) {x} : (x ∈ ⨅ i, p i) ↔ ∀ i, x ∈ p i := by
rw [← SetLike.mem_coe, iInf_coe, Set.mem_iInter]; rfl

theorem sInf_glb (S : Set (LieSubmodule R L M)) : IsGLB S (sInf S) := by
have h : ∀ {N N' : LieSubmodule R L M}, (N : Set M) ≤ N' ↔ N ≤ N' := fun {_ _} ↦ Iff.rfl
apply IsGLB.of_image h
Expand Down
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