algebra.lie.base_change | 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

9264b15e

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

36938f77

bump to nightly-2023-11-29-11

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

049e2cad

Diff

@@ -131,9 +131,9 @@ private theorem bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ =
   · intro z₁ z₂ h₁ h₂
     simp only [h₁, h₂, smul_add, add_lie]
 
-#print LieAlgebra.ExtendScalars.lieAlgebra /-
-instance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul := bracket_lie_smul R A L
-#align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.lieAlgebra
+#print LieAlgebra.ExtendScalars.instLieAlgebra /-
+instance instLieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul := bracket_lie_smul R A L
+#align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.instLieAlgebra
 -/
 
 end ExtendScalars

36938f77

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,8 +3,8 @@ 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.Algebra.RestrictScalars
-import Mathbin.Algebra.Lie.TensorProduct
+import Algebra.Algebra.RestrictScalars
+import Algebra.Lie.TensorProduct
 
 #align_import algebra.lie.base_change from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"

36938f77

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,15 +2,12 @@
 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.base_change
-! 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.Algebra.RestrictScalars
 import Mathbin.Algebra.Lie.TensorProduct
 
+#align_import algebra.lie.base_change from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
 /-!
 # Extension and restriction of scalars for Lie algebras

36938f77

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -61,10 +61,12 @@ instance : Bracket (A ⊗[R] L) (A ⊗[R] L) where bracket x y := bracket' R A L
 private theorem bracket_def (x y : A ⊗[R] L) : ⁅x, y⁆ = bracket' R A L x y :=
   rfl
 
+#print LieAlgebra.ExtendScalars.bracket_tmul /-
 @[simp]
 theorem bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := by
   rw [bracket_def, bracket'_tmul]
 #align lie_algebra.extend_scalars.bracket_tmul LieAlgebra.ExtendScalars.bracket_tmul
+-/
 
 private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 :=
   by
@@ -132,8 +134,10 @@ private theorem bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ =
   · intro z₁ z₂ h₁ h₂
     simp only [h₁, h₂, smul_add, add_lie]
 
+#print LieAlgebra.ExtendScalars.lieAlgebra /-
 instance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul := bracket_lie_smul R A L
 #align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.lieAlgebra
+-/
 
 end ExtendScalars
 
@@ -143,8 +147,6 @@ open RestrictScalars
 
 variable [h : LieRing L]
 
-include h
-
 instance : LieRing (RestrictScalars R A L) :=
   h

36938f77

bump to nightly-2023-06-07-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

831e78ce

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.base_change
-! leanprover-community/mathlib commit 9264b15ee696b7ca83f13c8ad67c83d6eb70b730
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Algebra.Lie.TensorProduct
 /-!
 # Extension and restriction of scalars for Lie algebras
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Lie algebras have a well-behaved theory of extension and restriction of scalars.
 
 ## Main definitions

9264b15e

bump to nightly-2023-06-05-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725

f4629da2

Diff

@@ -147,12 +147,14 @@ instance : LieRing (RestrictScalars R A L) :=
 
 variable [CommRing A] [LieAlgebra A L]
 
+#print LieAlgebra.RestrictScalars.lieAlgebra /-
 instance lieAlgebra [CommRing R] [Algebra R A] : LieAlgebra R (RestrictScalars R A L)
     where lie_smul t x y :=
     (lie_smul (algebraMap R A t) (RestrictScalars.addEquiv R A L x)
         (RestrictScalars.addEquiv R A L y) :
       _)
 #align lie_algebra.restrict_scalars.lie_algebra LieAlgebra.RestrictScalars.lieAlgebra
+-/
 
 end RestrictScalars

9264b15e

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -29,7 +29,7 @@ lie ring, lie algebra, extension of scalars, restriction of scalars, base change
 
 universe u v w w₁ w₂ w₃
 
-open TensorProduct
+open scoped TensorProduct
 
 variable (R : Type u) (A : Type w) (L : Type v)

9264b15e

bump to nightly-2023-05-28-04

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

6797a637

Diff

@@ -76,8 +76,7 @@ private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 :=
         zero_add, add_zero, add_comm, this]
     apply z₁.induction_on
     · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
-    · intro a₁ l₁
-      apply z₂.induction_on
+    · intro a₁ l₁; apply z₂.induction_on
       · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
       · intro a₂ l₂
         simp only [← lie_skew l₂ l₁, mul_comm a₁ a₂, TensorProduct.tmul_neg, bracket'_tmul,
@@ -98,8 +97,7 @@ private theorem bracket_leibniz_lie (x y z : A ⊗[R] L) : ⁅x, ⁅y, z⁆⁆ =
     · intro a₂ l₂
       apply z.induction_on
       · simp only [LinearMap.map_zero, add_zero]
-      · intro a₃ l₃
-        simp only [bracket'_tmul]
+      · intro a₃ l₃; simp only [bracket'_tmul]
         rw [mul_left_comm a₂ a₁ a₃, mul_assoc, leibniz_lie, TensorProduct.tmul_add]
       · intro u₁ u₂ h₁ h₂
         simp only [add_add_add_comm, h₁, h₂, LinearMap.map_add]
@@ -119,8 +117,7 @@ private theorem bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ =
   by
   apply x.induction_on
   · simp only [zero_lie, smul_zero]
-  · intro a₁ l₁
-    apply y.induction_on
+  · intro a₁ l₁; apply y.induction_on
     · simp only [lie_zero, smul_zero]
     · intro a₂ l₂
       simp only [bracket_def, bracket', TensorProduct.smul_tmul', mul_left_comm a₁ a a₂,

9264b15e

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -48,18 +48,15 @@ private def bracket' : A ⊗[R] L →ₗ[R] A ⊗[R] L →ₗ[R] A ⊗[R] L :=
   TensorProduct.curry <|
     TensorProduct.map (LinearMap.mul' R _) (LieModule.toModuleHom R L L : L ⊗[R] L →ₗ[R] L) ∘ₗ
       ↑(TensorProduct.tensorTensorTensorComm R A L A L)
-#align lie_algebra.extend_scalars.bracket' lie_algebra.extend_scalars.bracket'
 
 @[simp]
 private theorem bracket'_tmul (s t : A) (x y : L) :
     bracket' R A L (s ⊗ₜ[R] x) (t ⊗ₜ[R] y) = (s * t) ⊗ₜ ⁅x, y⁆ := by simp [bracket']
-#align lie_algebra.extend_scalars.bracket'_tmul lie_algebra.extend_scalars.bracket'_tmul
 
 instance : Bracket (A ⊗[R] L) (A ⊗[R] L) where bracket x y := bracket' R A L x y
 
 private theorem bracket_def (x y : A ⊗[R] L) : ⁅x, y⁆ = bracket' R A L x y :=
   rfl
-#align lie_algebra.extend_scalars.bracket_def lie_algebra.extend_scalars.bracket_def
 
 @[simp]
 theorem bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := by
@@ -89,7 +86,6 @@ private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 :=
         simp only [hy₁, hy₂, add_add_add_comm, add_zero, LinearMap.add_apply, LinearMap.map_add]
     · intro y₁ y₂ hy₁ hy₂
       simp only [add_add_add_comm, hy₁, hy₂, add_zero, LinearMap.add_apply, LinearMap.map_add]
-#align lie_algebra.extend_scalars.bracket_lie_self lie_algebra.extend_scalars.bracket_lie_self
 
 private theorem bracket_leibniz_lie (x y z : A ⊗[R] L) : ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆ :=
   by
@@ -111,7 +107,6 @@ private theorem bracket_leibniz_lie (x y z : A ⊗[R] L) : ⁅x, ⁅y, z⁆⁆ =
       simp only [add_add_add_comm, h₁, h₂, LinearMap.add_apply, LinearMap.map_add]
   · intro u₁ u₂ h₁ h₂
     simp only [add_add_add_comm, h₁, h₂, LinearMap.add_apply, LinearMap.map_add]
-#align lie_algebra.extend_scalars.bracket_leibniz_lie lie_algebra.extend_scalars.bracket_leibniz_lie
 
 instance : LieRing (A ⊗[R] L)
     where
@@ -136,7 +131,6 @@ private theorem bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ =
       simp only [h₁, h₂, smul_add, lie_add]
   · intro z₁ z₂ h₁ h₂
     simp only [h₁, h₂, smul_add, add_lie]
-#align lie_algebra.extend_scalars.bracket_lie_smul lie_algebra.extend_scalars.bracket_lie_smul
 
 instance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul := bracket_lie_smul R A L
 #align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.lieAlgebra

9264b15e

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

9264b15e

Acl/reorg tensor product (#11282)

Move:

Mathlib/Algebra/Module/DirectLimitAndTensorProduct.lean to LinearAlgebra/TensorProduct/DirectLimit.lean
Mathlib/LinearAlgebra/TensorProduct to Mathlib/LinearAlgebra.TensorProduct.Basic.lean
Mathlib/RingTheory/TensorProduct to Mathlib/RingTheory/TensorProduct/Basic.lean.

This follows suggestions 1, 2, 3 of

https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Tensor.20Products.20of.20modules.20and.20rings/near/424605543

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

0ee5f823

Diff

@@ -6,7 +6,7 @@ Authors: Oliver Nash
 import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.Algebra.Lie.TensorProduct
 import Mathlib.LinearAlgebra.TensorProduct.Tower
-import Mathlib.RingTheory.TensorProduct
+import Mathlib.RingTheory.TensorProduct.Basic
 
 #align_import algebra.lie.base_change from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"

9264b15e

chore: prepare Lean version bump with explicit simp (#10999)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

38bce757

Diff

@@ -218,7 +218,7 @@ lemma baseChange_top : (⊤ : LieSubmodule R L M).baseChange A = ⊤ := by
 lemma lie_baseChange {I : LieIdeal R L} {N : LieSubmodule R L M} :
     ⁅I, N⁆.baseChange A = ⁅I.baseChange A, N.baseChange A⁆ := by
   set s : Set (A ⊗[R] M) := { m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ ⁅x, n⁆ = m}
-  have : (TensorProduct.mk R A M 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s := by ext; simp
+  have : (TensorProduct.mk R A M 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s := by ext; simp [s]
   rw [← coe_toSubmodule_eq_iff, coe_baseChange, lieIdeal_oper_eq_linear_span',
     Submodule.baseChange_span, this, lieIdeal_oper_eq_linear_span']
   refine le_antisymm (Submodule.span_mono ?_) (Submodule.span_le.mpr ?_)

9264b15e

chore: delete LinearMap.extendScalars which duplicates LinearMap.baseChange (#8617)

For consistency, we also rename Submodule.extendScalars to Submodule.baseChange and likewise for LieSubmodule.

632ea778

Diff

@@ -6,6 +6,7 @@ Authors: Oliver Nash
 import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.Algebra.Lie.TensorProduct
 import Mathlib.LinearAlgebra.TensorProduct.Tower
+import Mathlib.RingTheory.TensorProduct
 
 #align_import algebra.lie.base_change from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"
 
@@ -152,8 +153,8 @@ variable [CommRing R] [LieRing L] [LieAlgebra R L]
   [CommRing A] [Algebra R A]
 
 @[simp]
-lemma LieModule.toEndomorphism_extendScalars (x : L) :
-    toEndomorphism A (A ⊗[R] L) (A ⊗[R] M) (1 ⊗ₜ x) = (toEndomorphism R L M x).extendScalars A := by
+lemma LieModule.toEndomorphism_baseChange (x : L) :
+    toEndomorphism A (A ⊗[R] L) (A ⊗[R] M) (1 ⊗ₜ x) = (toEndomorphism R L M x).baseChange A := by
   ext; simp
 
 namespace LieSubmodule
@@ -164,9 +165,11 @@ open LieModule
 
 variable {R L M} in
 /-- If `A` is an `R`-algebra, any Lie submodule of a Lie module `M` with coefficients in `R` may be
-pushed forward to a Lie submodule of `A ⊗ M` with coefficients in `A`. -/
-def extendScalars : LieSubmodule A (A ⊗[R] L) (A ⊗[R] M) :=
-  { (N : Submodule R M).extendScalars A with
+pushed forward to a Lie submodule of `A ⊗ M` with coefficients in `A`.
+
+This "base change" operation is also known as "extension of scalars". -/
+def baseChange : LieSubmodule A (A ⊗[R] L) (A ⊗[R] M) :=
+  { (N : Submodule R M).baseChange A with
     lie_mem := by
       intro x m hm
       simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
@@ -176,56 +179,56 @@ def extendScalars : LieSubmodule A (A ⊗[R] L) (A ⊗[R] M) :=
       · change toEndomorphism A (A ⊗[R] L) (A ⊗[R] M) _ _ ∈ _
         simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum, map_smul, toEndomorphism_apply_apply]
         suffices ∀ n : (N : Submodule R M).map (TensorProduct.mk R A M 1),
-            ⁅a ⊗ₜ[R] y, (n : A ⊗[R] M)⁆ ∈ (N : Submodule R M).extendScalars A by
+            ⁅a ⊗ₜ[R] y, (n : A ⊗[R] M)⁆ ∈ (N : Submodule R M).baseChange A by
           exact Submodule.sum_mem _ fun n _ ↦ Submodule.smul_mem _ _ (this n)
         rintro ⟨-, ⟨n : M, hn : n ∈ N, rfl⟩⟩
-        exact Submodule.tmul_mem_extendScalars_of_mem _ (N.lie_mem hn)
+        exact Submodule.tmul_mem_baseChange_of_mem _ (N.lie_mem hn)
       · rw [add_lie]
-        exact ((N : Submodule R M).extendScalars A).add_mem hy hz }
+        exact ((N : Submodule R M).baseChange A).add_mem hy hz }
 
 @[simp]
-lemma coe_extendScalars :
-    (N.extendScalars A : Submodule A (A ⊗[R] M)) = (N : Submodule R M).extendScalars A :=
+lemma coe_baseChange :
+    (N.baseChange A : Submodule A (A ⊗[R] M)) = (N : Submodule R M).baseChange A :=
   rfl
 
 variable {N}
 
 variable {R A L M} in
-lemma tmul_mem_extendScalars_of_mem (a : A) {m : M} (hm : m ∈ N) :
-    a ⊗ₜ[R] m ∈ N.extendScalars A :=
-  (N : Submodule R M).tmul_mem_extendScalars_of_mem a hm
+lemma tmul_mem_baseChange_of_mem (a : A) {m : M} (hm : m ∈ N) :
+    a ⊗ₜ[R] m ∈ N.baseChange A :=
+  (N : Submodule R M).tmul_mem_baseChange_of_mem a hm
 
-lemma mem_extendScalars_iff {m : A ⊗[R] M} :
-    m ∈ N.extendScalars A ↔
+lemma mem_baseChange_iff {m : A ⊗[R] M} :
+    m ∈ N.baseChange A ↔
     m ∈ Submodule.span A ((N : Submodule R M).map (TensorProduct.mk R A M 1)) :=
   Iff.rfl
 
 @[simp]
-lemma extendScalars_bot : (⊥ : LieSubmodule R L M).extendScalars A = ⊥ := by
-  simp only [extendScalars, bot_coeSubmodule, Submodule.extendScalars_bot,
+lemma baseChange_bot : (⊥ : LieSubmodule R L M).baseChange A = ⊥ := by
+  simp only [baseChange, bot_coeSubmodule, Submodule.baseChange_bot,
     Submodule.bot_toAddSubmonoid]
   rfl
 
 @[simp]
-lemma extendScalars_top : (⊤ : LieSubmodule R L M).extendScalars A = ⊤ := by
-  simp only [extendScalars, top_coeSubmodule, Submodule.extendScalars_top,
+lemma baseChange_top : (⊤ : LieSubmodule R L M).baseChange A = ⊤ := by
+  simp only [baseChange, top_coeSubmodule, Submodule.baseChange_top,
     Submodule.bot_toAddSubmonoid]
   rfl
 
-lemma lie_extendScalars {I : LieIdeal R L} {N : LieSubmodule R L M} :
-    ⁅I, N⁆.extendScalars A = ⁅I.extendScalars A, N.extendScalars A⁆ := by
+lemma lie_baseChange {I : LieIdeal R L} {N : LieSubmodule R L M} :
+    ⁅I, N⁆.baseChange A = ⁅I.baseChange A, N.baseChange A⁆ := by
   set s : Set (A ⊗[R] M) := { m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ ⁅x, n⁆ = m}
   have : (TensorProduct.mk R A M 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s := by ext; simp
-  rw [← coe_toSubmodule_eq_iff, coe_extendScalars, lieIdeal_oper_eq_linear_span',
-    Submodule.extendScalars_span, this, lieIdeal_oper_eq_linear_span']
+  rw [← coe_toSubmodule_eq_iff, coe_baseChange, lieIdeal_oper_eq_linear_span',
+    Submodule.baseChange_span, this, lieIdeal_oper_eq_linear_span']
   refine le_antisymm (Submodule.span_mono ?_) (Submodule.span_le.mpr ?_)
   · rintro - ⟨x, hx, m, hm, rfl⟩
-    exact ⟨1 ⊗ₜ x, tmul_mem_extendScalars_of_mem 1 hx,
-           1 ⊗ₜ m, tmul_mem_extendScalars_of_mem 1 hm, by simp⟩
+    exact ⟨1 ⊗ₜ x, tmul_mem_baseChange_of_mem 1 hx,
+           1 ⊗ₜ m, tmul_mem_baseChange_of_mem 1 hm, by simp⟩
   · rintro - ⟨x, hx, m, hm, rfl⟩
     revert m
     apply Submodule.span_induction
-      (p := fun x' ↦ ∀ m' ∈ N.extendScalars A, ⁅x', m'⁆ ∈ Submodule.span A s) hx
+      (p := fun x' ↦ ∀ m' ∈ N.baseChange A, ⁅x', m'⁆ ∈ Submodule.span A s) hx
     · rintro _ ⟨y : L, hy : y ∈ I, rfl⟩ m hm
       apply Submodule.span_induction (p := fun m' ↦ ⁅(1 : A) ⊗ₜ[R] y, m'⁆ ∈ Submodule.span A s) hm
       · rintro - ⟨m', hm' : m' ∈ N, rfl⟩

9264b15e

feat: remove triangularizability assumption from proof that Cartan subalgebras are Abelian (#8583)

99875260

Diff

@@ -144,3 +144,109 @@ instance lieAlgebra [CommRing R] [Algebra R A] : LieAlgebra R (RestrictScalars R
 end RestrictScalars
 
 end LieAlgebra
+
+section ExtendScalars
+
+variable [CommRing R] [LieRing L] [LieAlgebra R L]
+  [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
+  [CommRing A] [Algebra R A]
+
+@[simp]
+lemma LieModule.toEndomorphism_extendScalars (x : L) :
+    toEndomorphism A (A ⊗[R] L) (A ⊗[R] M) (1 ⊗ₜ x) = (toEndomorphism R L M x).extendScalars A := by
+  ext; simp
+
+namespace LieSubmodule
+
+variable (N : LieSubmodule R L M)
+
+open LieModule
+
+variable {R L M} in
+/-- If `A` is an `R`-algebra, any Lie submodule of a Lie module `M` with coefficients in `R` may be
+pushed forward to a Lie submodule of `A ⊗ M` with coefficients in `A`. -/
+def extendScalars : LieSubmodule A (A ⊗[R] L) (A ⊗[R] M) :=
+  { (N : Submodule R M).extendScalars A with
+    lie_mem := by
+      intro x m hm
+      simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
+        Submodule.mem_toAddSubmonoid] at hm ⊢
+      obtain ⟨c, rfl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp hm
+      refine x.induction_on (by simp) (fun a y ↦ ?_) (fun y z hy hz ↦ ?_)
+      · change toEndomorphism A (A ⊗[R] L) (A ⊗[R] M) _ _ ∈ _
+        simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum, map_smul, toEndomorphism_apply_apply]
+        suffices ∀ n : (N : Submodule R M).map (TensorProduct.mk R A M 1),
+            ⁅a ⊗ₜ[R] y, (n : A ⊗[R] M)⁆ ∈ (N : Submodule R M).extendScalars A by
+          exact Submodule.sum_mem _ fun n _ ↦ Submodule.smul_mem _ _ (this n)
+        rintro ⟨-, ⟨n : M, hn : n ∈ N, rfl⟩⟩
+        exact Submodule.tmul_mem_extendScalars_of_mem _ (N.lie_mem hn)
+      · rw [add_lie]
+        exact ((N : Submodule R M).extendScalars A).add_mem hy hz }
+
+@[simp]
+lemma coe_extendScalars :
+    (N.extendScalars A : Submodule A (A ⊗[R] M)) = (N : Submodule R M).extendScalars A :=
+  rfl
+
+variable {N}
+
+variable {R A L M} in
+lemma tmul_mem_extendScalars_of_mem (a : A) {m : M} (hm : m ∈ N) :
+    a ⊗ₜ[R] m ∈ N.extendScalars A :=
+  (N : Submodule R M).tmul_mem_extendScalars_of_mem a hm
+
+lemma mem_extendScalars_iff {m : A ⊗[R] M} :
+    m ∈ N.extendScalars A ↔
+    m ∈ Submodule.span A ((N : Submodule R M).map (TensorProduct.mk R A M 1)) :=
+  Iff.rfl
+
+@[simp]
+lemma extendScalars_bot : (⊥ : LieSubmodule R L M).extendScalars A = ⊥ := by
+  simp only [extendScalars, bot_coeSubmodule, Submodule.extendScalars_bot,
+    Submodule.bot_toAddSubmonoid]
+  rfl
+
+@[simp]
+lemma extendScalars_top : (⊤ : LieSubmodule R L M).extendScalars A = ⊤ := by
+  simp only [extendScalars, top_coeSubmodule, Submodule.extendScalars_top,
+    Submodule.bot_toAddSubmonoid]
+  rfl
+
+lemma lie_extendScalars {I : LieIdeal R L} {N : LieSubmodule R L M} :
+    ⁅I, N⁆.extendScalars A = ⁅I.extendScalars A, N.extendScalars A⁆ := by
+  set s : Set (A ⊗[R] M) := { m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ ⁅x, n⁆ = m}
+  have : (TensorProduct.mk R A M 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s := by ext; simp
+  rw [← coe_toSubmodule_eq_iff, coe_extendScalars, lieIdeal_oper_eq_linear_span',
+    Submodule.extendScalars_span, this, lieIdeal_oper_eq_linear_span']
+  refine le_antisymm (Submodule.span_mono ?_) (Submodule.span_le.mpr ?_)
+  · rintro - ⟨x, hx, m, hm, rfl⟩
+    exact ⟨1 ⊗ₜ x, tmul_mem_extendScalars_of_mem 1 hx,
+           1 ⊗ₜ m, tmul_mem_extendScalars_of_mem 1 hm, by simp⟩
+  · rintro - ⟨x, hx, m, hm, rfl⟩
+    revert m
+    apply Submodule.span_induction
+      (p := fun x' ↦ ∀ m' ∈ N.extendScalars A, ⁅x', m'⁆ ∈ Submodule.span A s) hx
+    · rintro _ ⟨y : L, hy : y ∈ I, rfl⟩ m hm
+      apply Submodule.span_induction (p := fun m' ↦ ⁅(1 : A) ⊗ₜ[R] y, m'⁆ ∈ Submodule.span A s) hm
+      · rintro - ⟨m', hm' : m' ∈ N, rfl⟩
+        rw [TensorProduct.mk_apply, LieAlgebra.ExtendScalars.bracket_tmul, mul_one]
+        apply Submodule.subset_span
+        exact ⟨y, hy, m', hm', rfl⟩
+      · simp
+      · intro u v hu hv
+        rw [lie_add]
+        exact Submodule.add_mem _ hu hv
+      · intro a u hu
+        rw [lie_smul]
+        exact Submodule.smul_mem _ a hu
+    · simp
+    · intro x y hx hy m' hm'
+      rw [add_lie]
+      exact Submodule.add_mem _ (hx _ hm') (hy _ hm')
+    · intro a x hx m' hm'
+      rw [smul_lie]
+      exact Submodule.smul_mem _ a (hx _ hm')
+
+end LieSubmodule
+
+end ExtendScalars

9264b15e

feat: base change for Lie modules (#8546)

3c148363

Diff

@@ -10,13 +10,15 @@ import Mathlib.LinearAlgebra.TensorProduct.Tower
 #align_import algebra.lie.base_change from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"
 
 /-!
-# Extension and restriction of scalars for Lie algebras
+# Extension and restriction of scalars for Lie algebras and Lie modules
 
-Lie algebras have a well-behaved theory of extension and restriction of scalars.
+Lie algebras and their representations have a well-behaved theory of extension and restriction of
+scalars.
 
 ## Main definitions
 
- * `LieAlgebra.ExtendScalars.lieAlgebra`
+ * `LieAlgebra.ExtendScalars.instLieAlgebra`
+ * `LieAlgebra.ExtendScalars.instLieModule`
  * `LieAlgebra.RestrictScalars.lieAlgebra`
 
 ## Tags
@@ -26,36 +28,35 @@ lie ring, lie algebra, extension of scalars, restriction of scalars, base change
 
 suppress_compilation
 
-universe u v w w₁ w₂ w₃
-
 open scoped TensorProduct
 
-variable (R : Type u) (A : Type w) (L : Type v)
+variable (R A L M : Type*)
 
 namespace LieAlgebra
 
 namespace ExtendScalars
 
 variable [CommRing R] [CommRing A] [Algebra R A] [LieRing L] [LieAlgebra R L]
+  [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
 
 /-- The Lie bracket on the extension of a Lie algebra `L` over `R` by an algebra `A` over `R`. -/
-private def bracket' : A ⊗[R] L →ₗ[A] A ⊗[R] L →ₗ[A] A ⊗[R] L :=
+private def bracket' : A ⊗[R] L →ₗ[A] A ⊗[R] M →ₗ[A] A ⊗[R] M :=
   TensorProduct.curry <|
     TensorProduct.AlgebraTensorModule.map
-        (LinearMap.mul' A A) (LieModule.toModuleHom R L L : L ⊗[R] L →ₗ[R] L) ∘ₗ
-      (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R A A L A L).toLinearMap
+        (LinearMap.mul' A A) (LieModule.toModuleHom R L M : L ⊗[R] M →ₗ[R] M) ∘ₗ
+      (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R A A L A M).toLinearMap
 
 @[simp]
-private theorem bracket'_tmul (s t : A) (x y : L) :
-    bracket' R A L (s ⊗ₜ[R] x) (t ⊗ₜ[R] y) = (s * t) ⊗ₜ ⁅x, y⁆ := rfl
+private theorem bracket'_tmul (s t : A) (x : L) (m : M) :
+    bracket' R A L M (s ⊗ₜ[R] x) (t ⊗ₜ[R] m) = (s * t) ⊗ₜ ⁅x, m⁆ := rfl
 
-instance : Bracket (A ⊗[R] L) (A ⊗[R] L) where bracket x y := bracket' R A L x y
+instance : Bracket (A ⊗[R] L) (A ⊗[R] M) where bracket x m := bracket' R A L M x m
 
-private theorem bracket_def (x y : A ⊗[R] L) : ⁅x, y⁆ = bracket' R A L x y :=
+private theorem bracket_def (x : A ⊗[R] L) (m : A ⊗[R] M) : ⁅x, m⁆ = bracket' R A L M x m :=
   rfl
 
 @[simp]
-theorem bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := rfl
+theorem bracket_tmul (s t : A) (x : L) (y : M) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := rfl
 #align lie_algebra.extend_scalars.bracket_tmul LieAlgebra.ExtendScalars.bracket_tmul
 
 private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
@@ -65,7 +66,7 @@ private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
   · intro a l
     simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
   · intro z₁ z₂ h₁ h₂
-    suffices bracket' R A L z₁ z₂ + bracket' R A L z₂ z₁ = 0 by
+    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
       rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
         zero_add, add_zero, add_comm, this]
     refine' z₁.induction_on _ _ _
@@ -80,7 +81,7 @@ private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
     · intro y₁ y₂ hy₁ hy₂
       simp only [add_add_add_comm, hy₁, hy₂, add_zero, LinearMap.add_apply, LinearMap.map_add]
 
-private theorem bracket_leibniz_lie (x y z : A ⊗[R] L) :
+private theorem bracket_leibniz_lie (x y : A ⊗[R] L) (z : A ⊗[R] M) :
     ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆ := by
   -- Porting note: replaced some `simp`s by `rw`s to avoid raising heartbeats
   simp only [bracket_def]
@@ -101,17 +102,26 @@ private theorem bracket_leibniz_lie (x y z : A ⊗[R] L) :
       rw [map_add, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add, map_add,
         LinearMap.add_apply, h₁, h₂, add_add_add_comm]
   · intro u₁ u₂ h₁ h₂
-    rw [map_add, LinearMap.add_apply, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add,
-      LinearMap.add_apply, h₁, h₂, add_add_add_comm]
+    rw [map_add, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add, LinearMap.add_apply,
+      map_add, LinearMap.add_apply, h₁, h₂, add_add_add_comm]
 
-instance : LieRing (A ⊗[R] L) where
+instance instLieRing : LieRing (A ⊗[R] L) where
   add_lie x y z := by simp only [bracket_def, LinearMap.add_apply, LinearMap.map_add]
   lie_add x y z := by simp only [bracket_def, LinearMap.map_add]
   lie_self := bracket_lie_self R A L
-  leibniz_lie := bracket_leibniz_lie R A L
+  leibniz_lie := bracket_leibniz_lie R A L L
+
+instance instLieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul _a _x _y := map_smul _ _ _
+#align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.instLieAlgebra
+
+instance instLieRingModule : LieRingModule (A ⊗[R] L) (A ⊗[R] M) where
+  add_lie x y z := by simp only [bracket_def, LinearMap.add_apply, LinearMap.map_add]
+  lie_add x y z := by simp only [bracket_def, LinearMap.map_add]
+  leibniz_lie := bracket_leibniz_lie R A L M
 
-instance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul _a _x _y := map_smul _ _ _
-#align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.lieAlgebra
+instance instLieModule : LieModule A (A ⊗[R] L) (A ⊗[R] M) where
+  smul_lie t x m := by simp only [bracket_def, map_smul, LinearMap.smul_apply]
+  lie_smul t x m := map_smul _ _ _
 
 end ExtendScalars

9264b15e

feat: use suppress_compilation in tensor products (#7504)

More principled version of #7281.

9ff5b6f1

Diff

@@ -24,6 +24,7 @@ Lie algebras have a well-behaved theory of extension and restriction of scalars.
 lie ring, lie algebra, extension of scalars, restriction of scalars, base change
 -/
 
+suppress_compilation
 
 universe u v w w₁ w₂ w₃

9264b15e

refactor(Algebra/Lie/BaseChange): use new tensor product machinery (#6628)

485b736d

Diff

@@ -5,6 +5,7 @@ Authors: Oliver Nash
 -/
 import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.Algebra.Lie.TensorProduct
+import Mathlib.LinearAlgebra.TensorProduct.Tower
 
 #align_import algebra.lie.base_change from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"
 
@@ -36,19 +37,16 @@ namespace ExtendScalars
 
 variable [CommRing R] [CommRing A] [Algebra R A] [LieRing L] [LieAlgebra R L]
 
-/-- The Lie bracket on the extension of a Lie algebra `L` over `R` by an algebra `A` over `R`.
-
-In fact this bracket is fully `A`-bilinear but without a significant upgrade to our mixed-scalar
-support in the tensor product library, it is far easier to bootstrap like this, starting with the
-definition below. -/
-private def bracket' : A ⊗[R] L →ₗ[R] A ⊗[R] L →ₗ[R] A ⊗[R] L :=
+/-- The Lie bracket on the extension of a Lie algebra `L` over `R` by an algebra `A` over `R`. -/
+private def bracket' : A ⊗[R] L →ₗ[A] A ⊗[R] L →ₗ[A] A ⊗[R] L :=
   TensorProduct.curry <|
-    TensorProduct.map (LinearMap.mul' R _) (LieModule.toModuleHom R L L : L ⊗[R] L →ₗ[R] L) ∘ₗ
-      ↑(TensorProduct.tensorTensorTensorComm R A L A L)
+    TensorProduct.AlgebraTensorModule.map
+        (LinearMap.mul' A A) (LieModule.toModuleHom R L L : L ⊗[R] L →ₗ[R] L) ∘ₗ
+      (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R A A L A L).toLinearMap
 
 @[simp]
 private theorem bracket'_tmul (s t : A) (x y : L) :
-    bracket' R A L (s ⊗ₜ[R] x) (t ⊗ₜ[R] y) = (s * t) ⊗ₜ ⁅x, y⁆ := by simp [bracket']
+    bracket' R A L (s ⊗ₜ[R] x) (t ⊗ₜ[R] y) = (s * t) ⊗ₜ ⁅x, y⁆ := rfl
 
 instance : Bracket (A ⊗[R] L) (A ⊗[R] L) where bracket x y := bracket' R A L x y
 
@@ -56,8 +54,7 @@ private theorem bracket_def (x y : A ⊗[R] L) : ⁅x, y⁆ = bracket' R A L x y
   rfl
 
 @[simp]
-theorem bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := by
-  rw [bracket_def, bracket'_tmul]
+theorem bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := rfl
 #align lie_algebra.extend_scalars.bracket_tmul LieAlgebra.ExtendScalars.bracket_tmul
 
 private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
@@ -112,22 +109,7 @@ instance : LieRing (A ⊗[R] L) where
   lie_self := bracket_lie_self R A L
   leibniz_lie := bracket_leibniz_lie R A L
 
-private theorem bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ = a • ⁅x, y⁆ := by
-  refine' x.induction_on _ _ _
-  · simp only [zero_lie, smul_zero]
-  · intro a₁ l₁; refine' y.induction_on _ _ _
-    · simp only [lie_zero, smul_zero]
-    · intro a₂ l₂
-      simp only [bracket_def, bracket', TensorProduct.smul_tmul', mul_left_comm a₁ a a₂,
-        TensorProduct.curry_apply, LinearMap.mul'_apply, Algebra.id.smul_eq_mul,
-        Function.comp_apply, LinearEquiv.coe_coe, LinearMap.coe_comp, TensorProduct.map_tmul,
-        TensorProduct.tensorTensorTensorComm_tmul]
-    · intro z₁ z₂ h₁ h₂
-      simp only [h₁, h₂, smul_add, lie_add]
-  · intro z₁ z₂ h₁ h₂
-    simp only [h₁, h₂, smul_add, add_lie]
-
-instance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul := bracket_lie_smul R A L
+instance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul _a _x _y := map_smul _ _ _
 #align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.lieAlgebra
 
 end ExtendScalars

9264b15e

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

@@ -2,15 +2,12 @@
 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.base_change
-! leanprover-community/mathlib commit 9264b15ee696b7ca83f13c8ad67c83d6eb70b730
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.Algebra.Lie.TensorProduct
 
+#align_import algebra.lie.base_change from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"
+
 /-!
 # Extension and restriction of scalars for Lie algebras

9264b15e

feat: port Algebra.Lie.BaseChange (#4650)

010d3eea

`algebra.lie.base_change` ⟷ `Mathlib.Algebra.Lie.BaseChange`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 471

algebra.lie.base_change ⟷ Mathlib.Algebra.Lie.BaseChange

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 471

`algebra.lie.base_change` ⟷ `Mathlib.Algebra.Lie.BaseChange`