algebra.lie.of_associative | Mathlib porting status

Changes since the initial port

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.

Changes in mathlib3

f0f3d964

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

5c1efce1

bump to nightly-2024-03-17-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

22f01ce4

Diff

@@ -373,7 +373,7 @@ def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [A
     (A' : Subalgebra R A) : LieSubalgebra R A :=
   { A'.toSubmodule with
     lie_mem' := fun x y hx hy => by
-      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx ; change y ∈ A' at hy 
+      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx; change y ∈ A' at hy
       rw [LieRing.of_associative_ring_bracket]
       have hxy := A'.mul_mem hx hy
       have hyx := A'.mul_mem hy hx

5c1efce1

bump to nightly-2024-02-23-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

ac9421d6

Diff

@@ -171,16 +171,16 @@ def LieModule.ofAssociativeModule : LieModule R A M
 #align lie_module.of_associative_module LieModule.ofAssociativeModule
 -/
 
-#print Module.End.lieRingModule /-
-instance Module.End.lieRingModule : LieRingModule (Module.End R M) M :=
+#print Module.End.instLieRingModule /-
+instance Module.End.instLieRingModule : LieRingModule (Module.End R M) M :=
   LieRingModule.ofAssociativeModule
-#align module.End.lie_ring_module Module.End.lieRingModule
+#align module.End.lie_ring_module Module.End.instLieRingModule
 -/
 
-#print Module.End.lieModule /-
-instance Module.End.lieModule : LieModule R (Module.End R M) M :=
+#print Module.End.instLieModule /-
+instance Module.End.instLieModule : LieModule R (Module.End R M) M :=
   LieModule.ofAssociativeModule
-#align module.End.lie_module Module.End.lieModule
+#align module.End.lie_module Module.End.instLieModule
 -/
 
 end AssociativeRepresentation

5c1efce1

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,10 +3,10 @@ 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.Basic
-import Mathbin.Algebra.Lie.Subalgebra
-import Mathbin.Algebra.Lie.Submodule
-import Mathbin.Algebra.Algebra.Subalgebra.Basic
+import Algebra.Lie.Basic
+import Algebra.Lie.Subalgebra
+import Algebra.Lie.Submodule
+import Algebra.Algebra.Subalgebra.Basic
 
 #align_import algebra.lie.of_associative from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"

5c1efce1

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,17 +2,14 @@
 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.of_associative
-! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Lie.Basic
 import Mathbin.Algebra.Lie.Subalgebra
 import Mathbin.Algebra.Lie.Submodule
 import Mathbin.Algebra.Algebra.Subalgebra.Basic
 
+#align_import algebra.lie.of_associative from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
+
 /-!
 # Lie algebras of associative algebras

5c1efce1

bump to nightly-2023-06-20-01

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

9b351f8c

Diff

@@ -294,7 +294,7 @@ theorem LieAlgebra.ad_apply (x y : L) : LieAlgebra.ad R L x y = ⁅x, y⁆ :=
 #print LieModule.toEndomorphism_module_end /-
 @[simp]
 theorem LieModule.toEndomorphism_module_end :
-    LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext (g m); simp [lie_eq_smul]
+    LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext g m; simp [lie_eq_smul]
 #align lie_module.to_endomorphism_module_End LieModule.toEndomorphism_module_end
 -/
 
@@ -353,7 +353,7 @@ open LieAlgebra
 
 #print LieAlgebra.ad_eq_lmul_left_sub_lmul_right /-
 theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
-    (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by ext (a b);
+    (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by ext a b;
   simp [LieRing.of_associative_ring_bracket]
 #align lie_algebra.ad_eq_lmul_left_sub_lmul_right LieAlgebra.ad_eq_lmul_left_sub_lmul_right
 -/

5c1efce1

bump to nightly-2023-06-19-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

bbb90f4e

Diff

@@ -213,11 +213,11 @@ theorem toLieHom_coe : f.toLieHom = ↑f :=
   rfl
 #align alg_hom.to_lie_hom_coe AlgHom.toLieHom_coe
 
-#print AlgHom.coe_to_lieHom /-
+#print AlgHom.coe_toLieHom /-
 @[simp]
-theorem coe_to_lieHom : ((f : A →ₗ⁅R⁆ B) : A → B) = f :=
+theorem coe_toLieHom : ((f : A →ₗ⁅R⁆ B) : A → B) = f :=
   rfl
-#align alg_hom.coe_to_lie_hom AlgHom.coe_to_lieHom
+#align alg_hom.coe_to_lie_hom AlgHom.coe_toLieHom
 -/
 
 #print AlgHom.toLieHom_apply /-
@@ -226,24 +226,24 @@ theorem toLieHom_apply (x : A) : f.toLieHom x = f x :=
 #align alg_hom.to_lie_hom_apply AlgHom.toLieHom_apply
 -/
 
-#print AlgHom.to_lieHom_id /-
+#print AlgHom.toLieHom_id /-
 @[simp]
-theorem to_lieHom_id : (AlgHom.id R A : A →ₗ⁅R⁆ A) = LieHom.id :=
+theorem toLieHom_id : (AlgHom.id R A : A →ₗ⁅R⁆ A) = LieHom.id :=
   rfl
-#align alg_hom.to_lie_hom_id AlgHom.to_lieHom_id
+#align alg_hom.to_lie_hom_id AlgHom.toLieHom_id
 -/
 
-#print AlgHom.to_lieHom_comp /-
+#print AlgHom.toLieHom_comp /-
 @[simp]
-theorem to_lieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) :=
+theorem toLieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) :=
   rfl
-#align alg_hom.to_lie_hom_comp AlgHom.to_lieHom_comp
+#align alg_hom.to_lie_hom_comp AlgHom.toLieHom_comp
 -/
 
-#print AlgHom.to_lieHom_injective /-
-theorem to_lieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
+#print AlgHom.toLieHom_injective /-
+theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
   ext a; exact LieHom.congr_fun h a
-#align alg_hom.to_lie_hom_injective AlgHom.to_lieHom_injective
+#align alg_hom.to_lie_hom_injective AlgHom.toLieHom_injective
 -/
 
 end AlgHom

5c1efce1

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -54,19 +54,25 @@ ring is commutative. It is identically zero exactly when the ring is commutative
 instance (priority := 100) : Bracket A A :=
   ⟨fun x y => x * y - y * x⟩
 
+#print Ring.lie_def /-
 theorem lie_def (x y : A) : ⁅x, y⁆ = x * y - y * x :=
   rfl
 #align ring.lie_def Ring.lie_def
+-/
 
 end Ring
 
+#print commute_iff_lie_eq /-
 theorem commute_iff_lie_eq {x y : A} : Commute x y ↔ ⁅x, y⁆ = 0 :=
   sub_eq_zero.symm
 #align commute_iff_lie_eq commute_iff_lie_eq
+-/
 
+#print Commute.lie_eq /-
 theorem Commute.lie_eq {x y : A} (h : Commute x y) : ⁅x, y⁆ = 0 :=
   sub_eq_zero_of_eq h
 #align commute.lie_eq Commute.lie_eq
+-/
 
 namespace LieRing
 
@@ -85,14 +91,18 @@ instance (priority := 100) ofAssociativeRing : LieRing A
 #align lie_ring.of_associative_ring LieRing.ofAssociativeRing
 -/
 
+#print LieRing.of_associative_ring_bracket /-
 theorem of_associative_ring_bracket (x y : A) : ⁅x, y⁆ = x * y - y * x :=
   rfl
 #align lie_ring.of_associative_ring_bracket LieRing.of_associative_ring_bracket
+-/
 
+#print LieRing.lie_apply /-
 @[simp]
 theorem lie_apply {α : Type _} (f g : α → A) (a : α) : ⁅f, g⁆ a = ⁅f a, g a⁆ :=
   rfl
 #align lie_ring.lie_apply LieRing.lie_apply
+-/
 
 end LieRing
 
@@ -170,9 +180,11 @@ instance Module.End.lieRingModule : LieRingModule (Module.End R M) M :=
 #align module.End.lie_ring_module Module.End.lieRingModule
 -/
 
+#print Module.End.lieModule /-
 instance Module.End.lieModule : LieModule R (Module.End R M) M :=
   LieModule.ofAssociativeModule
 #align module.End.lie_module Module.End.lieModule
+-/
 
 end AssociativeRepresentation
 
@@ -201,14 +213,18 @@ theorem toLieHom_coe : f.toLieHom = ↑f :=
   rfl
 #align alg_hom.to_lie_hom_coe AlgHom.toLieHom_coe
 
+#print AlgHom.coe_to_lieHom /-
 @[simp]
 theorem coe_to_lieHom : ((f : A →ₗ⁅R⁆ B) : A → B) = f :=
   rfl
 #align alg_hom.coe_to_lie_hom AlgHom.coe_to_lieHom
+-/
 
+#print AlgHom.toLieHom_apply /-
 theorem toLieHom_apply (x : A) : f.toLieHom x = f x :=
   rfl
 #align alg_hom.to_lie_hom_apply AlgHom.toLieHom_apply
+-/
 
 #print AlgHom.to_lieHom_id /-
 @[simp]
@@ -244,6 +260,7 @@ variable [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M]
 
 variable [LieRingModule L M] [LieModule R L M]
 
+#print LieModule.toEndomorphism /-
 /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
 
 See also `lie_module.to_module_hom`. -/
@@ -258,32 +275,43 @@ def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M
   map_smul' t x := by ext m; apply smul_lie
   map_lie' x y := by ext m; apply lie_lie
 #align lie_module.to_endomorphism LieModule.toEndomorphism
+-/
 
+#print LieAlgebra.ad /-
 /-- The adjoint action of a Lie algebra on itself. -/
 def LieAlgebra.ad : L →ₗ⁅R⁆ Module.End R L :=
   LieModule.toEndomorphism R L L
 #align lie_algebra.ad LieAlgebra.ad
+-/
 
+#print LieAlgebra.ad_apply /-
 @[simp]
 theorem LieAlgebra.ad_apply (x y : L) : LieAlgebra.ad R L x y = ⁅x, y⁆ :=
   rfl
 #align lie_algebra.ad_apply LieAlgebra.ad_apply
+-/
 
+#print LieModule.toEndomorphism_module_end /-
 @[simp]
 theorem LieModule.toEndomorphism_module_end :
     LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext (g m); simp [lie_eq_smul]
 #align lie_module.to_endomorphism_module_End LieModule.toEndomorphism_module_end
+-/
 
+#print LieSubalgebra.toEndomorphism_eq /-
 theorem LieSubalgebra.toEndomorphism_eq (K : LieSubalgebra R L) {x : K} :
     LieModule.toEndomorphism R K M x = LieModule.toEndomorphism R L M x :=
   rfl
 #align lie_subalgebra.to_endomorphism_eq LieSubalgebra.toEndomorphism_eq
+-/
 
+#print LieSubalgebra.toEndomorphism_mk /-
 @[simp]
 theorem LieSubalgebra.toEndomorphism_mk (K : LieSubalgebra R L) {x : L} (hx : x ∈ K) :
     LieModule.toEndomorphism R K M ⟨x, hx⟩ = LieModule.toEndomorphism R L M x :=
   rfl
 #align lie_subalgebra.to_endomorphism_mk LieSubalgebra.toEndomorphism_mk
+-/
 
 variable {R L M}
 
@@ -323,10 +351,12 @@ end LieSubmodule
 
 open LieAlgebra
 
+#print LieAlgebra.ad_eq_lmul_left_sub_lmul_right /-
 theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
     (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by ext (a b);
   simp [LieRing.of_associative_ring_bracket]
 #align lie_algebra.ad_eq_lmul_left_sub_lmul_right LieAlgebra.ad_eq_lmul_left_sub_lmul_right
+-/
 
 #print LieSubalgebra.ad_comp_incl_eq /-
 theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
@@ -362,6 +392,7 @@ variable [CommRing R] [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [M
 
 variable (e : M₁ ≃ₗ[R] M₂)
 
+#print LinearEquiv.lieConj /-
 /-- A linear equivalence of two modules induces a Lie algebra equivalence of their endomorphisms. -/
 def lieConj : Module.End R M₁ ≃ₗ⁅R⁆ Module.End R M₂ :=
   { e.conj with
@@ -370,16 +401,21 @@ def lieConj : Module.End R M₁ ≃ₗ⁅R⁆ Module.End R M₂ :=
         simp only [LieRing.of_associative_ring_bracket, LinearMap.mul_eq_comp, e.conj_comp,
           LinearEquiv.map_sub] }
 #align linear_equiv.lie_conj LinearEquiv.lieConj
+-/
 
+#print LinearEquiv.lieConj_apply /-
 @[simp]
 theorem lieConj_apply (f : Module.End R M₁) : e.lieConj f = e.conj f :=
   rfl
 #align linear_equiv.lie_conj_apply LinearEquiv.lieConj_apply
+-/
 
+#print LinearEquiv.lieConj_symm /-
 @[simp]
 theorem lieConj_symm : e.lieConj.symm = e.symm.lieConj :=
   rfl
 #align linear_equiv.lie_conj_symm LinearEquiv.lieConj_symm
+-/
 
 end LinearEquiv
 
@@ -400,15 +436,19 @@ def toLieEquiv : A₁ ≃ₗ⁅R⁆ A₂ :=
 #align alg_equiv.to_lie_equiv AlgEquiv.toLieEquiv
 -/
 
+#print AlgEquiv.toLieEquiv_apply /-
 @[simp]
 theorem toLieEquiv_apply (x : A₁) : e.toLieEquiv x = e x :=
   rfl
 #align alg_equiv.to_lie_equiv_apply AlgEquiv.toLieEquiv_apply
+-/
 
+#print AlgEquiv.toLieEquiv_symm_apply /-
 @[simp]
 theorem toLieEquiv_symm_apply (x : A₂) : e.toLieEquiv.symm x = e.symm x :=
   rfl
 #align alg_equiv.to_lie_equiv_symm_apply AlgEquiv.toLieEquiv_symm_apply
+-/
 
 end AlgEquiv

5c1efce1

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -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.of_associative
-! leanprover-community/mathlib commit f0f3d964763ecd0090c9eb3ae0d15871d08781c4
+! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Algebra.Algebra.Subalgebra.Basic
 /-!
 # Lie algebras of associative algebras
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines the Lie algebra structure that arises on an associative algebra via the ring
 commutator.
 
@@ -67,6 +70,7 @@ theorem Commute.lie_eq {x y : A} (h : Commute x y) : ⁅x, y⁆ = 0 :=
 
 namespace LieRing
 
+#print LieRing.ofAssociativeRing /-
 /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
 instance (priority := 100) ofAssociativeRing : LieRing A
     where
@@ -79,6 +83,7 @@ instance (priority := 100) ofAssociativeRing : LieRing A
   lie_self := by simp only [Ring.lie_def, forall_const, sub_self]
   leibniz_lie x y z := by repeat' rw [Ring.lie_def]; noncomm_ring
 #align lie_ring.of_associative_ring LieRing.ofAssociativeRing
+-/
 
 theorem of_associative_ring_bracket (x y : A) : ⁅x, y⁆ = x * y - y * x :=
   rfl
@@ -95,6 +100,7 @@ section AssociativeModule
 
 variable {M : Type w} [AddCommGroup M] [Module A M]
 
+#print LieRingModule.ofAssociativeModule /-
 /-- We can regard a module over an associative ring `A` as a Lie ring module over `A` with Lie
 bracket equal to its ring commutator.
 
@@ -113,12 +119,15 @@ def LieRingModule.ofAssociativeModule : LieRingModule A M
   lie_add := smul_add
   leibniz_lie := by simp [LieRing.of_associative_ring_bracket, sub_smul, mul_smul, sub_add_cancel]
 #align lie_ring_module.of_associative_module LieRingModule.ofAssociativeModule
+-/
 
 attribute [local instance] LieRingModule.ofAssociativeModule
 
+#print lie_eq_smul /-
 theorem lie_eq_smul (a : A) (m : M) : ⁅a, m⁆ = a • m :=
   rfl
 #align lie_eq_smul lie_eq_smul
+-/
 
 end AssociativeModule
 
@@ -126,6 +135,7 @@ section LieAlgebra
 
 variable {R : Type u} [CommRing R] [Algebra R A]
 
+#print LieAlgebra.ofAssociativeAlgebra /-
 /-- An associative algebra gives rise to a Lie algebra by taking the bracket to be the ring
 commutator. -/
 instance (priority := 100) LieAlgebra.ofAssociativeAlgebra : LieAlgebra R A
@@ -133,6 +143,7 @@ instance (priority := 100) LieAlgebra.ofAssociativeAlgebra : LieAlgebra R A
     rw [LieRing.of_associative_ring_bracket, LieRing.of_associative_ring_bracket,
       Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_sub]
 #align lie_algebra.of_associative_algebra LieAlgebra.ofAssociativeAlgebra
+-/
 
 attribute [local instance] LieRingModule.ofAssociativeModule
 
@@ -140,6 +151,7 @@ section AssociativeRepresentation
 
 variable {M : Type w} [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M]
 
+#print LieModule.ofAssociativeModule /-
 /-- A representation of an associative algebra `A` is also a representation of `A`, regarded as a
 Lie algebra via the ring commutator.
 
@@ -150,10 +162,13 @@ def LieModule.ofAssociativeModule : LieModule R A M
   smul_lie := smul_assoc
   lie_smul := smul_algebra_smul_comm
 #align lie_module.of_associative_module LieModule.ofAssociativeModule
+-/
 
+#print Module.End.lieRingModule /-
 instance Module.End.lieRingModule : LieRingModule (Module.End R M) M :=
   LieRingModule.ofAssociativeModule
 #align module.End.lie_ring_module Module.End.lieRingModule
+-/
 
 instance Module.End.lieModule : LieModule R (Module.End R M) M :=
   LieModule.ofAssociativeModule
@@ -167,6 +182,7 @@ variable {B : Type w} {C : Type w₁} [Ring B] [Ring C] [Algebra R B] [Algebra R
 
 variable (f : A →ₐ[R] B) (g : B →ₐ[R] C)
 
+#print AlgHom.toLieHom /-
 /-- The map `of_associative_algebra` associating a Lie algebra to an associative algebra is
 functorial. -/
 def toLieHom : A →ₗ⁅R⁆ B :=
@@ -175,6 +191,7 @@ def toLieHom : A →ₗ⁅R⁆ B :=
       show f ⁅x, y⁆ = ⁅f x, f y⁆ by
         simp only [LieRing.of_associative_ring_bracket, AlgHom.map_sub, AlgHom.map_mul] }
 #align alg_hom.to_lie_hom AlgHom.toLieHom
+-/
 
 instance : Coe (A →ₐ[R] B) (A →ₗ⁅R⁆ B) :=
   ⟨toLieHom⟩
@@ -193,19 +210,25 @@ theorem toLieHom_apply (x : A) : f.toLieHom x = f x :=
   rfl
 #align alg_hom.to_lie_hom_apply AlgHom.toLieHom_apply
 
+#print AlgHom.to_lieHom_id /-
 @[simp]
 theorem to_lieHom_id : (AlgHom.id R A : A →ₗ⁅R⁆ A) = LieHom.id :=
   rfl
 #align alg_hom.to_lie_hom_id AlgHom.to_lieHom_id
+-/
 
+#print AlgHom.to_lieHom_comp /-
 @[simp]
 theorem to_lieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) :=
   rfl
 #align alg_hom.to_lie_hom_comp AlgHom.to_lieHom_comp
+-/
 
+#print AlgHom.to_lieHom_injective /-
 theorem to_lieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
   ext a; exact LieHom.congr_fun h a
 #align alg_hom.to_lie_hom_injective AlgHom.to_lieHom_injective
+-/
 
 end AlgHom
 
@@ -270,25 +293,31 @@ open LieModule
 
 variable {N : LieSubmodule R L M} {x : L}
 
+#print LieSubmodule.coe_map_toEndomorphism_le /-
 theorem coe_map_toEndomorphism_le :
     (N : Submodule R M).map (LieModule.toEndomorphism R L M x) ≤ N :=
   by
   rintro n ⟨m, hm, rfl⟩
   exact N.lie_mem hm
 #align lie_submodule.coe_map_to_endomorphism_le LieSubmodule.coe_map_toEndomorphism_le
+-/
 
 variable (N x)
 
+#print LieSubmodule.toEndomorphism_comp_subtype_mem /-
 theorem toEndomorphism_comp_subtype_mem (m : M) (hm : m ∈ (N : Submodule R M)) :
     (toEndomorphism R L M x).comp (N : Submodule R M).Subtype ⟨m, hm⟩ ∈ (N : Submodule R M) := by
   simpa using N.lie_mem hm
 #align lie_submodule.to_endomorphism_comp_subtype_mem LieSubmodule.toEndomorphism_comp_subtype_mem
+-/
 
+#print LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism /-
 @[simp]
 theorem toEndomorphism_restrict_eq_toEndomorphism (h := N.toEndomorphism_comp_subtype_mem x) :
     (toEndomorphism R L M x).restrict h = toEndomorphism R L N x := by ext;
   simp [LinearMap.restrict_apply]
 #align lie_submodule.to_endomorphism_restrict_eq_to_endomorphism LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism
+-/
 
 end LieSubmodule
 
@@ -299,6 +328,7 @@ theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra
   simp [LieRing.of_associative_ring_bracket]
 #align lie_algebra.ad_eq_lmul_left_sub_lmul_right LieAlgebra.ad_eq_lmul_left_sub_lmul_right
 
+#print LieSubalgebra.ad_comp_incl_eq /-
 theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
     (ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) :=
   by
@@ -306,9 +336,11 @@ theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
   simp only [ad_apply, LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, LinearMap.coe_comp,
     LieSubalgebra.coe_bracket, Function.comp_apply]
 #align lie_subalgebra.ad_comp_incl_eq LieSubalgebra.ad_comp_incl_eq
+-/
 
 end AdjointAction
 
+#print lieSubalgebraOfSubalgebra /-
 /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
 def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
     (A' : Subalgebra R A) : LieSubalgebra R A :=
@@ -320,6 +352,7 @@ def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [A
       have hyx := A'.mul_mem hy hx
       exact Submodule.sub_mem A'.to_submodule hxy hyx }
 #align lie_subalgebra_of_subalgebra lieSubalgebraOfSubalgebra
+-/
 
 namespace LinearEquiv
 
@@ -358,12 +391,14 @@ variable [CommRing R] [Ring A₁] [Ring A₂] [Algebra R A₁] [Algebra R A₂]
 
 variable (e : A₁ ≃ₐ[R] A₂)
 
+#print AlgEquiv.toLieEquiv /-
 /-- An equivalence of associative algebras is an equivalence of associated Lie algebras. -/
 def toLieEquiv : A₁ ≃ₗ⁅R⁆ A₂ :=
   { e.toLinearEquiv with
     toFun := e.toFun
     map_lie' := fun x y => by simp [LieRing.of_associative_ring_bracket] }
 #align alg_equiv.to_lie_equiv AlgEquiv.toLieEquiv
+-/
 
 @[simp]
 theorem toLieEquiv_apply (x : A₁) : e.toLieEquiv x = e x :=

f0f3d964

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -314,7 +314,7 @@ def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [A
     (A' : Subalgebra R A) : LieSubalgebra R A :=
   { A'.toSubmodule with
     lie_mem' := fun x y hx hy => by
-      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx; change y ∈ A' at hy
+      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx ; change y ∈ A' at hy 
       rw [LieRing.of_associative_ring_bracket]
       have hxy := A'.mul_mem hx hy
       have hyx := A'.mul_mem hy hx

f0f3d964

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -77,9 +77,7 @@ instance (priority := 100) ofAssociativeRing : LieRing A
     simp only [Ring.lie_def, right_distrib, left_distrib, sub_eq_add_neg, add_comm, add_left_comm,
       forall_const, eq_self_iff_true, neg_add_rev]
   lie_self := by simp only [Ring.lie_def, forall_const, sub_self]
-  leibniz_lie x y z := by
-    repeat' rw [Ring.lie_def]
-    noncomm_ring
+  leibniz_lie x y z := by repeat' rw [Ring.lie_def]; noncomm_ring
 #align lie_ring.of_associative_ring LieRing.ofAssociativeRing
 
 theorem of_associative_ring_bracket (x y : A) : ⁅x, y⁆ = x * y - y * x :=
@@ -205,10 +203,8 @@ theorem to_lieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C
   rfl
 #align alg_hom.to_lie_hom_comp AlgHom.to_lieHom_comp
 
-theorem to_lieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g :=
-  by
-  ext a
-  exact LieHom.congr_fun h a
+theorem to_lieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
+  ext a; exact LieHom.congr_fun h a
 #align alg_hom.to_lie_hom_injective AlgHom.to_lieHom_injective
 
 end AlgHom
@@ -235,15 +231,9 @@ def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M
     { toFun := fun m => ⁅x, m⁆
       map_add' := lie_add x
       map_smul' := fun t => lie_smul t x }
-  map_add' x y := by
-    ext m
-    apply add_lie
-  map_smul' t x := by
-    ext m
-    apply smul_lie
-  map_lie' x y := by
-    ext m
-    apply lie_lie
+  map_add' x y := by ext m; apply add_lie
+  map_smul' t x := by ext m; apply smul_lie
+  map_lie' x y := by ext m; apply lie_lie
 #align lie_module.to_endomorphism LieModule.toEndomorphism
 
 /-- The adjoint action of a Lie algebra on itself. -/
@@ -258,10 +248,7 @@ theorem LieAlgebra.ad_apply (x y : L) : LieAlgebra.ad R L x y = ⁅x, y⁆ :=
 
 @[simp]
 theorem LieModule.toEndomorphism_module_end :
-    LieModule.toEndomorphism R (Module.End R M) M = LieHom.id :=
-  by
-  ext (g m)
-  simp [lie_eq_smul]
+    LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext (g m); simp [lie_eq_smul]
 #align lie_module.to_endomorphism_module_End LieModule.toEndomorphism_module_end
 
 theorem LieSubalgebra.toEndomorphism_eq (K : LieSubalgebra R L) {x : K} :
@@ -299,9 +286,7 @@ theorem toEndomorphism_comp_subtype_mem (m : M) (hm : m ∈ (N : Submodule R M))
 
 @[simp]
 theorem toEndomorphism_restrict_eq_toEndomorphism (h := N.toEndomorphism_comp_subtype_mem x) :
-    (toEndomorphism R L M x).restrict h = toEndomorphism R L N x :=
-  by
-  ext
+    (toEndomorphism R L M x).restrict h = toEndomorphism R L N x := by ext;
   simp [LinearMap.restrict_apply]
 #align lie_submodule.to_endomorphism_restrict_eq_to_endomorphism LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism
 
@@ -310,9 +295,7 @@ end LieSubmodule
 open LieAlgebra
 
 theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
-    (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R :=
-  by
-  ext (a b)
+    (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by ext (a b);
   simp [LieRing.of_associative_ring_bracket]
 #align lie_algebra.ad_eq_lmul_left_sub_lmul_right LieAlgebra.ad_eq_lmul_left_sub_lmul_right
 
@@ -331,9 +314,7 @@ def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [A
     (A' : Subalgebra R A) : LieSubalgebra R A :=
   { A'.toSubmodule with
     lie_mem' := fun x y hx hy => by
-      change ⁅x, y⁆ ∈ A'
-      change x ∈ A' at hx
-      change y ∈ A' at hy
+      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx; change y ∈ A' at hy
       rw [LieRing.of_associative_ring_bracket]
       have hxy := A'.mul_mem hx hy
       have hyx := A'.mul_mem hy hx

f0f3d964

bump to nightly-2023-02-27-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

feafc24f

Diff

@@ -320,7 +320,7 @@ theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
     (ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) :=
   by
   ext y
-  simp only [ad_apply, LieHom.coe_to_linearMap, LieSubalgebra.coe_incl, LinearMap.coe_comp,
+  simp only [ad_apply, LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, LinearMap.coe_comp,
     LieSubalgebra.coe_bracket, Function.comp_apply]
 #align lie_subalgebra.ad_comp_incl_eq LieSubalgebra.ad_comp_incl_eq

f0f3d964

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

f0f3d964

chore: remove redundant LinearEquiv.map_neg/sub (#12330)

These are redundant with _root_.{map_neg,map_sub}.

ae4dd74c

Diff

@@ -377,7 +377,7 @@ def lieConj : Module.End R M₁ ≃ₗ⁅R⁆ Module.End R M₂ :=
     map_lie' := fun {f g} =>
       show e.conj ⁅f, g⁆ = ⁅e.conj f, e.conj g⁆ by
         simp only [LieRing.of_associative_ring_bracket, LinearMap.mul_eq_comp, e.conj_comp,
-          LinearEquiv.map_sub] }
+          map_sub] }
 #align linear_equiv.lie_conj LinearEquiv.lieConj
 
 @[simp]

f0f3d964

feat(Algebra/Lie): prove that IsKilling transfers by isomorphisms (#12008)

This proves isKilling_of_equiv, which states that if a Lie algebra is isomorphic to a Killing Lie algebra, then it is Killing too.

This is a step towards the proof that all derivations over a finite dimensional semisimple Lie algebra are inner (see this thread). Indeed, the proof that I have formalized relies on the fact that such a Lie algebra L is isomorphic to ad(L), from which I want to infer that ad(L) is Killing.

76ab9daf

Diff

@@ -418,3 +418,21 @@ theorem toLieEquiv_symm_apply (x : A₂) : e.toLieEquiv.symm x = e.symm x :=
 #align alg_equiv.to_lie_equiv_symm_apply AlgEquiv.toLieEquiv_symm_apply
 
 end AlgEquiv
+
+namespace LieAlgebra
+
+variable {R L L' : Type*} [CommRing R]
+  [LieRing L] [LieAlgebra R L]
+  [LieRing L'] [LieAlgebra R L']
+
+open LieEquiv
+
+/-- Given an equivalence `e` of Lie algebras from `L` to `L'`, and an element `x : L`, the conjugate
+of the endomorphism `ad(x)` of `L` by `e` is the endomorphism `ad(e x)` of `L'`. -/
+@[simp]
+lemma conj_ad_apply (e : L ≃ₗ⁅R⁆ L') (x : L) : LinearEquiv.conj e (ad R L x) = ad R L' (e x) := by
+  ext y'
+  rw [LinearEquiv.conj_apply_apply, ad_apply, ad_apply, coe_to_linearEquiv, map_lie,
+    ← coe_to_linearEquiv, LinearEquiv.apply_symm_apply]
+
+end LieAlgebra

f0f3d964

change the order of operation in zsmulRec and nsmulRec (#11451)

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

pow_succ and pow_succ' have switched their meanings.
Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

9a3feeae

Diff

@@ -271,8 +271,8 @@ lemma LieModule.toEndomorphism_pow_lie (x y : L) (z : M) (n : ℕ) :
   | succ n ih =>
     rw [Finset.sum_antidiagonal_choose_succ_nsmul
       (fun i j ↦ ⁅((ad R L x) ^ i) y, ((φ x) ^ j) z⁆) n]
-    simp only [pow_succ, LinearMap.mul_apply, ih, map_sum, map_nsmul, toEndomorphism_lie, nsmul_add,
-      sum_add_distrib]
+    simp only [pow_succ', LinearMap.mul_apply, ih, map_sum, map_nsmul,
+      toEndomorphism_lie, nsmul_add, sum_add_distrib]
     rw [add_comm, add_left_cancel_iff, sum_congr rfl]
     rintro ⟨i, j⟩ hij
     rw [mem_antidiagonal] at hij

f0f3d964

chore(*): remove empty lines between variable statements (#11418)

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)

75c86f04

Diff

@@ -136,7 +136,6 @@ end AssociativeRepresentation
 namespace AlgHom
 
 variable {B : Type w} {C : Type w₁} [Ring B] [Ring C] [Algebra R B] [Algebra R C]
-
 variable (f : A →ₐ[R] B) (g : B →ₐ[R] C)
 
 /-- The map `ofAssociativeAlgebra` associating a Lie algebra to an associative algebra is
@@ -188,9 +187,7 @@ end OfAssociative
 section AdjointAction
 
 variable (R : Type u) (L : Type v) (M : Type w)
-
 variable [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M]
-
 variable [LieRingModule L M] [LieModule R L M]
 
 /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
@@ -371,9 +368,7 @@ def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [A
 namespace LinearEquiv
 
 variable {R : Type u} {M₁ : Type v} {M₂ : Type w}
-
 variable [CommRing R] [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂]
-
 variable (e : M₁ ≃ₗ[R] M₂)
 
 /-- A linear equivalence of two modules induces a Lie algebra equivalence of their endomorphisms. -/
@@ -400,9 +395,7 @@ end LinearEquiv
 namespace AlgEquiv
 
 variable {R : Type u} {A₁ : Type v} {A₂ : Type w}
-
 variable [CommRing R] [Ring A₁] [Ring A₂] [Algebra R A₁] [Algebra R A₂]
-
 variable (e : A₁ ≃ₐ[R] A₂)
 
 /-- An equivalence of associative algebras is an equivalence of associated Lie algebras. -/

f0f3d964

feat: basic theory of chains of roots / weights of Lie algebras / modules (#10548)

4ae8a4dd

Diff

@@ -121,13 +121,15 @@ theorem LieModule.ofAssociativeModule : LieModule R A M where
   lie_smul := smul_algebra_smul_comm
 #align lie_module.of_associative_module LieModule.ofAssociativeModule
 
-instance Module.End.lieRingModule : LieRingModule (Module.End R M) M :=
+instance Module.End.instLieRingModule : LieRingModule (Module.End R M) M :=
   LieRingModule.ofAssociativeModule
-#align module.End.lie_ring_module Module.End.lieRingModule
+#align module.End.lie_ring_module Module.End.instLieRingModule
 
-instance Module.End.lieModule : LieModule R (Module.End R M) M :=
+instance Module.End.instLieModule : LieModule R (Module.End R M) M :=
   LieModule.ofAssociativeModule
-#align module.End.lie_module Module.End.lieModule
+#align module.End.lie_module Module.End.instLieModule
+
+@[simp] lemma Module.End.lie_apply (f : Module.End R M) (m : M) : ⁅f, m⁆ = f m := rfl
 
 end AssociativeRepresentation

f0f3d964

feat: add lemma LinearMap.trace_lie and move Bracket instance (#10584)

It is useful to have access to this bracket definition without having to import a bunch of Lie algebra theory.

d43978d8

Diff

@@ -41,27 +41,6 @@ section OfAssociative
 
 variable {A : Type v} [Ring A]
 
-namespace Ring
-
-/-- The bracket operation for rings is the ring commutator, which captures the extent to which a
-ring is commutative. It is identically zero exactly when the ring is commutative. -/
-instance (priority := 100) instBracket : Bracket A A :=
-  ⟨fun x y => x * y - y * x⟩
-
-theorem lie_def (x y : A) : ⁅x, y⁆ = x * y - y * x :=
-  rfl
-#align ring.lie_def Ring.lie_def
-
-end Ring
-
-theorem commute_iff_lie_eq {x y : A} : Commute x y ↔ ⁅x, y⁆ = 0 :=
-  sub_eq_zero.symm
-#align commute_iff_lie_eq commute_iff_lie_eq
-
-theorem Commute.lie_eq {x y : A} (h : Commute x y) : ⁅x, y⁆ = 0 :=
-  sub_eq_zero_of_eq h
-#align commute.lie_eq Commute.lie_eq
-
 namespace LieRing
 
 /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/

f0f3d964

feat(Algebra/Lie): Engel subalgebras (#10158)

Progress toward https://github.com/leanprover-community/mathlib4/issues/10065

a0cb47af

Diff

@@ -252,6 +252,62 @@ theorem LieSubalgebra.toEndomorphism_mk (K : LieSubalgebra R L) {x : L} (hx : x
   rfl
 #align lie_subalgebra.to_endomorphism_mk LieSubalgebra.toEndomorphism_mk
 
+section
+
+open BigOperators LieAlgebra LieModule
+
+lemma LieSubmodule.coe_toEndomorphism (N : LieSubmodule R L M) (x : L) (y : N) :
+    (toEndomorphism R L N x y : M) = toEndomorphism R L M x y := rfl
+
+lemma LieSubmodule.coe_toEndomorphism_pow (N : LieSubmodule R L M) (x : L) (y : N) (n : ℕ) :
+    ((toEndomorphism R L N x ^ n) y : M) = (toEndomorphism R L M x ^ n) y := by
+  induction n generalizing y with
+  | zero => rfl
+  | succ n ih => simp only [pow_succ', LinearMap.mul_apply, ih, LieSubmodule.coe_toEndomorphism]
+
+lemma LieSubalgebra.coe_ad (H : LieSubalgebra R L) (x y : H) :
+    (ad R H x y : L) = ad R L x y := rfl
+
+lemma LieSubalgebra.coe_ad_pow (H : LieSubalgebra R L) (x y : H) (n : ℕ) :
+    ((ad R H x ^ n) y : L) = (ad R L x ^ n) y :=
+  LieSubmodule.coe_toEndomorphism_pow R H L H.toLieSubmodule x y n
+
+variable {L M}
+
+local notation "φ" => LieModule.toEndomorphism R L M
+
+lemma LieModule.toEndomorphism_lie (x y : L) (z : M) :
+    (φ x) ⁅y, z⁆ = ⁅ad R L x y, z⁆ + ⁅y, φ x z⁆ := by
+  simp
+
+lemma LieAlgebra.ad_lie (x y z : L) :
+    (ad R L x) ⁅y, z⁆ = ⁅ad R L x y, z⁆ + ⁅y, ad R L x z⁆ :=
+  toEndomorphism_lie _ x y z
+
+open Finset in
+lemma LieModule.toEndomorphism_pow_lie (x y : L) (z : M) (n : ℕ) :
+    ((φ x) ^ n) ⁅y, z⁆ =
+      ∑ ij in antidiagonal n, n.choose ij.1 • ⁅((ad R L x) ^ ij.1) y, ((φ x) ^ ij.2) z⁆ := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_antidiagonal_choose_succ_nsmul
+      (fun i j ↦ ⁅((ad R L x) ^ i) y, ((φ x) ^ j) z⁆) n]
+    simp only [pow_succ, LinearMap.mul_apply, ih, map_sum, map_nsmul, toEndomorphism_lie, nsmul_add,
+      sum_add_distrib]
+    rw [add_comm, add_left_cancel_iff, sum_congr rfl]
+    rintro ⟨i, j⟩ hij
+    rw [mem_antidiagonal] at hij
+    rw [Nat.choose_symm_of_eq_add hij.symm]
+
+open Finset in
+lemma LieAlgebra.ad_pow_lie (x y z : L) (n : ℕ) :
+    ((ad R L x) ^ n) ⁅y, z⁆ =
+      ∑ ij in antidiagonal n, n.choose ij.1 • ⁅((ad R L x) ^ ij.1) y, ((ad R L x) ^ ij.2) z⁆ :=
+  toEndomorphism_pow_lie _ x y z n
+
+end
+
 variable {R L M}
 
 namespace LieModule

f0f3d964

feat: weight space decomposition of Lie modules (#7963)

87c833e0

Diff

@@ -273,7 +273,7 @@ end LieModule
 
 namespace LieSubmodule
 
-open LieModule
+open LieModule Set
 
 variable {N : LieSubmodule R L M} {x : L}
 
@@ -296,6 +296,11 @@ theorem toEndomorphism_restrict_eq_toEndomorphism (h := N.toEndomorphism_comp_su
   ext; simp [LinearMap.restrict_apply]
 #align lie_submodule.to_endomorphism_restrict_eq_to_endomorphism LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism
 
+lemma mapsTo_pow_toEndomorphism_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
+    MapsTo ((toEndomorphism R L M x - algebraMap R (Module.End R M) φ) ^ k) N N := by
+  rw [LinearMap.coe_pow]
+  exact MapsTo.iterate (fun m hm ↦ N.sub_mem (N.lie_mem hm) (N.smul_mem _ hm)) k
+
 end LieSubmodule
 
 open LieAlgebra

f0f3d964

feat: miscellaneous Lie algebra lemmas (#7782)

9eb25ab7

Diff

@@ -45,7 +45,7 @@ namespace Ring
 
 /-- The bracket operation for rings is the ring commutator, which captures the extent to which a
 ring is commutative. It is identically zero exactly when the ring is commutative. -/
-instance (priority := 100) : Bracket A A :=
+instance (priority := 100) instBracket : Bracket A A :=
   ⟨fun x y => x * y - y * x⟩
 
 theorem lie_def (x y : A) : ⁅x, y⁆ = x * y - y * x :=

f0f3d964

feat: various Lie algebra definitions / lemmas (#7507)

These are all motivated by a result I've proved but I believe they make sense in their own right so I have split them out in the hopes of simplifying review.

8cdd933d

Diff

@@ -254,6 +254,23 @@ theorem LieSubalgebra.toEndomorphism_mk (K : LieSubalgebra R L) {x : L} (hx : x
 
 variable {R L M}
 
+namespace LieModule
+
+variable {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂]
+  (f : M →ₗ⁅R,L⁆ M₂) (k : ℕ) (x : L)
+
+lemma toEndomorphism_pow_comp_lieHom :
+    (toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k := by
+  apply LinearMap.commute_pow_left_of_commute
+  ext
+  simp
+
+lemma toEndomorphism_pow_apply_map (m : M) :
+    (toEndomorphism R L M₂ x ^ k) (f m) = f ((toEndomorphism R L M x ^ k) m) :=
+  LinearMap.congr_fun (toEndomorphism_pow_comp_lieHom f k x) m
+
+end LieModule
+
 namespace LieSubmodule
 
 open LieModule

f0f3d964

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -78,7 +78,7 @@ theorem of_associative_ring_bracket (x y : A) : ⁅x, y⁆ = x * y - y * x :=
 #align lie_ring.of_associative_ring_bracket LieRing.of_associative_ring_bracket
 
 @[simp]
-theorem lie_apply {α : Type _} (f g : α → A) (a : α) : ⁅f, g⁆ a = ⁅f a, g a⁆ :=
+theorem lie_apply {α : Type*} (f g : α → A) (a : α) : ⁅f, g⁆ a = ⁅f a, g a⁆ :=
   rfl
 #align lie_ring.lie_apply LieRing.lie_apply

f0f3d964

feat: Linter that checks that Prop classes are Props (#6148)

930dd29d

Diff

@@ -137,7 +137,7 @@ Lie algebra via the ring commutator.
 
 See the comment at `LieRingModule.ofAssociativeModule` for why the possibility `M = A` means
 this cannot be a global instance. -/
-def LieModule.ofAssociativeModule : LieModule R A M where
+theorem LieModule.ofAssociativeModule : LieModule R A M where
   smul_lie := smul_assoc
   lie_smul := smul_algebra_smul_comm
 #align lie_module.of_associative_module LieModule.ofAssociativeModule

f0f3d964

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,17 +2,14 @@
 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.of_associative
-! leanprover-community/mathlib commit f0f3d964763ecd0090c9eb3ae0d15871d08781c4
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.Lie.Basic
 import Mathlib.Algebra.Lie.Subalgebra
 import Mathlib.Algebra.Lie.Submodule
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 
+#align_import algebra.lie.of_associative from "leanprover-community/mathlib"@"f0f3d964763ecd0090c9eb3ae0d15871d08781c4"
+
 /-!
 # Lie algebras of associative algebras

f0f3d964

chore: remove occurrences of semicolon after space (#5713)

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

c795bd35

Diff

@@ -305,7 +305,7 @@ def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [A
     (A' : Subalgebra R A) : LieSubalgebra R A :=
   { Subalgebra.toSubmodule A' with
     lie_mem' := fun {x y} hx hy => by
-      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx ; change y ∈ A' at hy
+      change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx; change y ∈ A' at hy
       rw [LieRing.of_associative_ring_bracket]
       have hxy := A'.mul_mem hx hy
       have hyx := A'.mul_mem hy hx

f0f3d964

chore: remove superfluous parentheses in calls to ext (#5258)

Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Pol'tta / Miyahara Kō <pol_tta@outlook.jp> Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

3d6731dc

Diff

@@ -241,7 +241,7 @@ theorem LieAlgebra.ad_apply (x y : L) : LieAlgebra.ad R L x y = ⁅x, y⁆ :=
 
 @[simp]
 theorem LieModule.toEndomorphism_module_end :
-    LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext (g m); simp [lie_eq_smul]
+    LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext g m; simp [lie_eq_smul]
 #align lie_module.to_endomorphism_module_End LieModule.toEndomorphism_module_end
 
 theorem LieSubalgebra.toEndomorphism_eq (K : LieSubalgebra R L) {x : K} :
@@ -288,7 +288,7 @@ open LieAlgebra
 
 theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
     (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by
-  ext (a b); simp [LieRing.of_associative_ring_bracket]
+  ext a b; simp [LieRing.of_associative_ring_bracket]
 #align lie_algebra.ad_eq_lmul_left_sub_lmul_right LieAlgebra.ad_eq_lmul_left_sub_lmul_right
 
 theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :

f0f3d964

chore: tidy various files (#5233)

697ac1d3

Diff

@@ -171,35 +171,35 @@ def toLieHom : A →ₗ⁅R⁆ B :=
 instance : Coe (A →ₐ[R] B) (A →ₗ⁅R⁆ B) :=
   ⟨toLieHom⟩
 
-/-- Porting note: is a syntactic tautology
+/- Porting note: is a syntactic tautology
 @[simp]
 theorem toLieHom_coe : f.toLieHom = ↑f :=
   rfl
-#align alg_hom.to_lie_hom_coe AlgHom.toLieHom_coe
 -/
+#noalign alg_hom.to_lie_hom_coe
 
 @[simp]
-theorem coe_to_lieHom : ((f : A →ₗ⁅R⁆ B) : A → B) = f :=
+theorem coe_toLieHom : ((f : A →ₗ⁅R⁆ B) : A → B) = f :=
   rfl
-#align alg_hom.coe_to_lie_hom AlgHom.coe_to_lieHom
+#align alg_hom.coe_to_lie_hom AlgHom.coe_toLieHom
 
 theorem toLieHom_apply (x : A) : f.toLieHom x = f x :=
   rfl
 #align alg_hom.to_lie_hom_apply AlgHom.toLieHom_apply
 
 @[simp]
-theorem to_lieHom_id : (AlgHom.id R A : A →ₗ⁅R⁆ A) = LieHom.id :=
+theorem toLieHom_id : (AlgHom.id R A : A →ₗ⁅R⁆ A) = LieHom.id :=
   rfl
-#align alg_hom.to_lie_hom_id AlgHom.to_lieHom_id
+#align alg_hom.to_lie_hom_id AlgHom.toLieHom_id
 
 @[simp]
-theorem to_lieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) :=
+theorem toLieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) :=
   rfl
-#align alg_hom.to_lie_hom_comp AlgHom.to_lieHom_comp
+#align alg_hom.to_lie_hom_comp AlgHom.toLieHom_comp
 
-theorem to_lieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
+theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
   ext a; exact LieHom.congr_fun h a
-#align alg_hom.to_lie_hom_injective AlgHom.to_lieHom_injective
+#align alg_hom.to_lie_hom_injective AlgHom.toLieHom_injective
 
 end AlgHom

f0f3d964

feat: port Algebra.Lie.OfAssociative (#4617)

00687379

`algebra.lie.of_associative` ⟷ `Mathlib.Algebra.Lie.OfAssociative`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 467

algebra.lie.of_associative ⟷ Mathlib.Algebra.Lie.OfAssociative

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 467

`algebra.lie.of_associative` ⟷ `Mathlib.Algebra.Lie.OfAssociative`