algebra.lie.semisimple
⟷
Mathlib.Algebra.Lie.Semisimple
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -92,7 +92,7 @@ theorem isSemisimple_iff_no_abelian_ideals :
#print LieAlgebra.center_eq_bot_of_semisimple /-
@[simp]
theorem center_eq_bot_of_semisimple [h : IsSemisimple R L] : center R L = ⊥ := by
- rw [is_semisimple_iff_no_abelian_ideals] at h ; apply h; infer_instance
+ rw [is_semisimple_iff_no_abelian_ideals] at h; apply h; infer_instance
#align lie_algebra.center_eq_bot_of_semisimple LieAlgebra.center_eq_bot_of_semisimple
-/
@@ -104,7 +104,7 @@ instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisim
intro I hI
obtain @⟨⟨h₁⟩, h₂⟩ := id h
by_contra contra
- rw [h₁ I contra, lie_abelian_iff_equiv_lie_abelian LieIdeal.topEquiv] at hI
+ rw [h₁ I contra, lie_abelian_iff_equiv_lie_abelian LieIdeal.topEquiv] at hI
exact h₂ hI
#align lie_algebra.is_semisimple_of_is_simple LieAlgebra.isSemisimpleOfIsSimple
-/
@@ -114,7 +114,7 @@ instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisim
theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbelian L] :
Subsingleton L :=
by
- rw [is_lie_abelian_iff_center_eq_top R L, center_eq_bot_of_semisimple] at h
+ rw [is_lie_abelian_iff_center_eq_top R L, center_eq_bot_of_semisimple] at h
exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h)
#align lie_algebra.subsingleton_of_semisimple_lie_abelian LieAlgebra.subsingleton_of_semisimple_lie_abelian
-/
@@ -136,7 +136,7 @@ theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
by
constructor
· rintro h₁ I h₂
- rw [lie_ideal.solvable_iff_le_radical] at h₂
+ rw [lie_ideal.solvable_iff_le_radical] at h₂
exact (LieIdeal.inclusion_injective h₂).IsLieAbelian h₁
· intro h; apply h; infer_instance
#align lie_algebra.abelian_radical_iff_solvable_is_abelian LieAlgebra.abelian_radical_iff_solvable_is_abelian
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -137,7 +137,7 @@ theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
constructor
· rintro h₁ I h₂
rw [lie_ideal.solvable_iff_le_radical] at h₂
- exact (LieIdeal.homOfLe_injective h₂).IsLieAbelian h₁
+ exact (LieIdeal.inclusion_injective h₂).IsLieAbelian h₁
· intro h; apply h; infer_instance
#align lie_algebra.abelian_radical_iff_solvable_is_abelian LieAlgebra.abelian_radical_iff_solvable_is_abelian
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
-import Mathbin.Algebra.Lie.Solvable
+import Algebra.Lie.Solvable
#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module algebra.lie.semisimple
-! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Algebra.Lie.Solvable
+#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
/-!
# Semisimple Lie algebras
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -112,6 +112,7 @@ instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisim
#align lie_algebra.is_semisimple_of_is_simple LieAlgebra.isSemisimpleOfIsSimple
-/
+#print LieAlgebra.subsingleton_of_semisimple_lie_abelian /-
/-- A semisimple Abelian Lie algebra is trivial. -/
theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbelian L] :
Subsingleton L :=
@@ -119,6 +120,7 @@ theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbel
rw [is_lie_abelian_iff_center_eq_top R L, center_eq_bot_of_semisimple] at h
exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h)
#align lie_algebra.subsingleton_of_semisimple_lie_abelian LieAlgebra.subsingleton_of_semisimple_lie_abelian
+-/
#print LieAlgebra.abelian_radical_of_semisimple /-
theorem abelian_radical_of_semisimple [IsSemisimple R L] : IsLieAbelian (radical R L) := by
mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
! This file was ported from Lean 3 source module algebra.lie.semisimple
-! leanprover-community/mathlib commit 356447fe00e75e54777321045cdff7c9ea212e60
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -13,6 +13,9 @@ import Mathbin.Algebra.Lie.Solvable
/-!
# Semisimple Lie algebras
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
The famous Cartan-Dynkin-Killing classification of semisimple Lie algebras renders them one of the
most important classes of Lie algebras. In this file we define simple and semisimple Lie algebras
and prove some basic related results.
mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725
@@ -34,12 +34,14 @@ lie algebra, radical, simple, semisimple
universe u v w w₁ w₂
+#print LieModule.IsIrreducible /-
/-- A Lie module is irreducible if it is zero or its only non-trivial Lie submodule is itself. -/
class LieModule.IsIrreducible (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L]
[LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Prop where
Irreducible : ∀ N : LieSubmodule R L M, N ≠ ⊥ → N = ⊤
#align lie_module.is_irreducible LieModule.IsIrreducible
+-/
namespace LieAlgebra
@@ -47,12 +49,15 @@ variable (R : Type u) (L : Type v)
variable [CommRing R] [LieRing L] [LieAlgebra R L]
+#print LieAlgebra.IsSimple /-
/-- A Lie algebra is simple if it is irreducible as a Lie module over itself via the adjoint
action, and it is non-Abelian. -/
class IsSimple extends LieModule.IsIrreducible R L L : Prop where
non_abelian : ¬IsLieAbelian L
#align lie_algebra.is_simple LieAlgebra.IsSimple
+-/
+#print LieAlgebra.IsSemisimple /-
/-- A semisimple Lie algebra is one with trivial radical.
Note that the label 'semisimple' is apparently not universally agreed
@@ -63,12 +68,16 @@ zero. -/
class IsSemisimple : Prop where
semisimple : radical R L = ⊥
#align lie_algebra.is_semisimple LieAlgebra.IsSemisimple
+-/
+#print LieAlgebra.isSemisimple_iff_no_solvable_ideals /-
theorem isSemisimple_iff_no_solvable_ideals :
IsSemisimple R L ↔ ∀ I : LieIdeal R L, IsSolvable R I → I = ⊥ :=
⟨fun h => sSup_eq_bot.mp h.semisimple, fun h => ⟨sSup_eq_bot.mpr h⟩⟩
#align lie_algebra.is_semisimple_iff_no_solvable_ideals LieAlgebra.isSemisimple_iff_no_solvable_ideals
+-/
+#print LieAlgebra.isSemisimple_iff_no_abelian_ideals /-
theorem isSemisimple_iff_no_abelian_ideals :
IsSemisimple R L ↔ ∀ I : LieIdeal R L, IsLieAbelian I → I = ⊥ :=
by
@@ -78,12 +87,16 @@ theorem isSemisimple_iff_no_abelian_ideals :
· haveI : IsSolvable R I := h₂; rw [← abelian_of_solvable_ideal_eq_bot_iff]; apply h₁
exact abelian_derived_abelian_of_ideal I
#align lie_algebra.is_semisimple_iff_no_abelian_ideals LieAlgebra.isSemisimple_iff_no_abelian_ideals
+-/
+#print LieAlgebra.center_eq_bot_of_semisimple /-
@[simp]
theorem center_eq_bot_of_semisimple [h : IsSemisimple R L] : center R L = ⊥ := by
rw [is_semisimple_iff_no_abelian_ideals] at h ; apply h; infer_instance
#align lie_algebra.center_eq_bot_of_semisimple LieAlgebra.center_eq_bot_of_semisimple
+-/
+#print LieAlgebra.isSemisimpleOfIsSimple /-
/-- A simple Lie algebra is semisimple. -/
instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisimple R L :=
by
@@ -94,6 +107,7 @@ instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisim
rw [h₁ I contra, lie_abelian_iff_equiv_lie_abelian LieIdeal.topEquiv] at hI
exact h₂ hI
#align lie_algebra.is_semisimple_of_is_simple LieAlgebra.isSemisimpleOfIsSimple
+-/
/-- A semisimple Abelian Lie algebra is trivial. -/
theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbelian L] :
@@ -103,10 +117,13 @@ theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbel
exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h)
#align lie_algebra.subsingleton_of_semisimple_lie_abelian LieAlgebra.subsingleton_of_semisimple_lie_abelian
+#print LieAlgebra.abelian_radical_of_semisimple /-
theorem abelian_radical_of_semisimple [IsSemisimple R L] : IsLieAbelian (radical R L) := by
rw [is_semisimple.semisimple]; exact is_lie_abelian_bot R L
#align lie_algebra.abelian_radical_of_semisimple LieAlgebra.abelian_radical_of_semisimple
+-/
+#print LieAlgebra.abelian_radical_iff_solvable_is_abelian /-
/-- The two properties shown to be equivalent here are possible definitions for a Lie algebra
to be reductive.
@@ -121,9 +138,12 @@ theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
exact (LieIdeal.homOfLe_injective h₂).IsLieAbelian h₁
· intro h; apply h; infer_instance
#align lie_algebra.abelian_radical_iff_solvable_is_abelian LieAlgebra.abelian_radical_iff_solvable_is_abelian
+-/
+#print LieAlgebra.ad_ker_eq_bot_of_semisimple /-
theorem ad_ker_eq_bot_of_semisimple [IsSemisimple R L] : (ad R L).ker = ⊥ := by simp
#align lie_algebra.ad_ker_eq_bot_of_semisimple LieAlgebra.ad_ker_eq_bot_of_semisimple
+-/
end LieAlgebra
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -36,7 +36,8 @@ universe u v w w₁ w₂
/-- A Lie module is irreducible if it is zero or its only non-trivial Lie submodule is itself. -/
class LieModule.IsIrreducible (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L]
- [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] : Prop where
+ [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
+ Prop where
Irreducible : ∀ N : LieSubmodule R L M, N ≠ ⊥ → N = ⊤
#align lie_module.is_irreducible LieModule.IsIrreducible
@@ -80,7 +81,7 @@ theorem isSemisimple_iff_no_abelian_ideals :
@[simp]
theorem center_eq_bot_of_semisimple [h : IsSemisimple R L] : center R L = ⊥ := by
- rw [is_semisimple_iff_no_abelian_ideals] at h; apply h; infer_instance
+ rw [is_semisimple_iff_no_abelian_ideals] at h ; apply h; infer_instance
#align lie_algebra.center_eq_bot_of_semisimple LieAlgebra.center_eq_bot_of_semisimple
/-- A simple Lie algebra is semisimple. -/
@@ -90,7 +91,7 @@ instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisim
intro I hI
obtain @⟨⟨h₁⟩, h₂⟩ := id h
by_contra contra
- rw [h₁ I contra, lie_abelian_iff_equiv_lie_abelian LieIdeal.topEquiv] at hI
+ rw [h₁ I contra, lie_abelian_iff_equiv_lie_abelian LieIdeal.topEquiv] at hI
exact h₂ hI
#align lie_algebra.is_semisimple_of_is_simple LieAlgebra.isSemisimpleOfIsSimple
@@ -98,7 +99,7 @@ instance (priority := 100) isSemisimpleOfIsSimple [h : IsSimple R L] : IsSemisim
theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbelian L] :
Subsingleton L :=
by
- rw [is_lie_abelian_iff_center_eq_top R L, center_eq_bot_of_semisimple] at h
+ rw [is_lie_abelian_iff_center_eq_top R L, center_eq_bot_of_semisimple] at h
exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h)
#align lie_algebra.subsingleton_of_semisimple_lie_abelian LieAlgebra.subsingleton_of_semisimple_lie_abelian
@@ -116,7 +117,7 @@ theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
by
constructor
· rintro h₁ I h₂
- rw [lie_ideal.solvable_iff_le_radical] at h₂
+ rw [lie_ideal.solvable_iff_le_radical] at h₂
exact (LieIdeal.homOfLe_injective h₂).IsLieAbelian h₁
· intro h; apply h; infer_instance
#align lie_algebra.abelian_radical_iff_solvable_is_abelian LieAlgebra.abelian_radical_iff_solvable_is_abelian
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -73,21 +73,14 @@ theorem isSemisimple_iff_no_abelian_ideals :
by
rw [is_semisimple_iff_no_solvable_ideals]
constructor <;> intro h₁ I h₂
- · haveI : IsLieAbelian I := h₂
- apply h₁
- exact LieAlgebra.ofAbelianIsSolvable R I
- · haveI : IsSolvable R I := h₂
- rw [← abelian_of_solvable_ideal_eq_bot_iff]
- apply h₁
+ · haveI : IsLieAbelian I := h₂; apply h₁; exact LieAlgebra.ofAbelianIsSolvable R I
+ · haveI : IsSolvable R I := h₂; rw [← abelian_of_solvable_ideal_eq_bot_iff]; apply h₁
exact abelian_derived_abelian_of_ideal I
#align lie_algebra.is_semisimple_iff_no_abelian_ideals LieAlgebra.isSemisimple_iff_no_abelian_ideals
@[simp]
-theorem center_eq_bot_of_semisimple [h : IsSemisimple R L] : center R L = ⊥ :=
- by
- rw [is_semisimple_iff_no_abelian_ideals] at h
- apply h
- infer_instance
+theorem center_eq_bot_of_semisimple [h : IsSemisimple R L] : center R L = ⊥ := by
+ rw [is_semisimple_iff_no_abelian_ideals] at h; apply h; infer_instance
#align lie_algebra.center_eq_bot_of_semisimple LieAlgebra.center_eq_bot_of_semisimple
/-- A simple Lie algebra is semisimple. -/
@@ -109,10 +102,8 @@ theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbel
exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h)
#align lie_algebra.subsingleton_of_semisimple_lie_abelian LieAlgebra.subsingleton_of_semisimple_lie_abelian
-theorem abelian_radical_of_semisimple [IsSemisimple R L] : IsLieAbelian (radical R L) :=
- by
- rw [is_semisimple.semisimple]
- exact is_lie_abelian_bot R L
+theorem abelian_radical_of_semisimple [IsSemisimple R L] : IsLieAbelian (radical R L) := by
+ rw [is_semisimple.semisimple]; exact is_lie_abelian_bot R L
#align lie_algebra.abelian_radical_of_semisimple LieAlgebra.abelian_radical_of_semisimple
/-- The two properties shown to be equivalent here are possible definitions for a Lie algebra
@@ -127,9 +118,7 @@ theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
· rintro h₁ I h₂
rw [lie_ideal.solvable_iff_le_radical] at h₂
exact (LieIdeal.homOfLe_injective h₂).IsLieAbelian h₁
- · intro h
- apply h
- infer_instance
+ · intro h; apply h; infer_instance
#align lie_algebra.abelian_radical_iff_solvable_is_abelian LieAlgebra.abelian_radical_iff_solvable_is_abelian
theorem ad_ker_eq_bot_of_semisimple [IsSemisimple R L] : (ad R L).ker = ⊥ := by simp
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -65,7 +65,7 @@ class IsSemisimple : Prop where
theorem isSemisimple_iff_no_solvable_ideals :
IsSemisimple R L ↔ ∀ I : LieIdeal R L, IsSolvable R I → I = ⊥ :=
- ⟨fun h => supₛ_eq_bot.mp h.semisimple, fun h => ⟨supₛ_eq_bot.mpr h⟩⟩
+ ⟨fun h => sSup_eq_bot.mp h.semisimple, fun h => ⟨sSup_eq_bot.mpr h⟩⟩
#align lie_algebra.is_semisimple_iff_no_solvable_ideals LieAlgebra.isSemisimple_iff_no_solvable_ideals
theorem isSemisimple_iff_no_abelian_ideals :
mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25
@@ -76,7 +76,7 @@ theorem isSemisimple_iff_no_abelian_ideals :
· haveI : IsLieAbelian I := h₂
apply h₁
exact LieAlgebra.ofAbelianIsSolvable R I
- · haveI : is_solvable R I := h₂
+ · haveI : IsSolvable R I := h₂
rw [← abelian_of_solvable_ideal_eq_bot_iff]
apply h₁
exact abelian_derived_abelian_of_ideal I
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
@@ -41,7 +41,6 @@ class LieModule.IsIrreducible (R : Type u) (L : Type v) (M : Type w) [CommRing R
namespace LieAlgebra
variable (R : Type u) (L : Type v)
-
variable [CommRing R] [LieRing L] [LieAlgebra R L]
/-- A Lie algebra is simple if it is irreducible as a Lie module over itself via the adjoint
Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.
@@ -98,7 +98,7 @@ theorem subsingleton_of_semisimple_lie_abelian [IsSemisimple R L] [h : IsLieAbel
#align lie_algebra.subsingleton_of_semisimple_lie_abelian LieAlgebra.subsingleton_of_semisimple_lie_abelian
theorem abelian_radical_of_semisimple [IsSemisimple R L] : IsLieAbelian (radical R L) := by
- rw [IsSemisimple.semisimple]; exact isLieAbelian_bot R L
+ rw [IsSemisimple.semisimple]; infer_instance
#align lie_algebra.abelian_radical_of_semisimple LieAlgebra.abelian_radical_of_semisimple
/-- The two properties shown to be equivalent here are possible definitions for a Lie algebra
Submodule.ofLe
to Submodule.inclusion
(#8470)
This matches Set.inclusion
, Subring.inclusion
, Subalgebra.inclusion
, etc.
Also renames the homOfLe
spellings in Algebra/Lie
to match.
Note that we leave LieSubalgebra.ofLe
, as this is a completely different statement!
As requested by @alreadydone.
@@ -111,7 +111,7 @@ theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
constructor
· rintro h₁ I h₂
rw [LieIdeal.solvable_iff_le_radical] at h₂
- exact (LieIdeal.homOfLe_injective h₂).isLieAbelian h₁
+ exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁
· intro h; apply h; infer_instance
#align lie_algebra.abelian_radical_iff_solvable_is_abelian LieAlgebra.abelian_radical_iff_solvable_is_abelian
@@ -2,14 +2,11 @@
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module algebra.lie.semisimple
-! leanprover-community/mathlib commit 356447fe00e75e54777321045cdff7c9ea212e60
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Algebra.Lie.Solvable
+#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"356447fe00e75e54777321045cdff7c9ea212e60"
+
/-!
# Semisimple Lie algebras
The unported dependencies are