algebra.lie.character
⟷
Mathlib.Algebra.Lie.Character
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
@@ -57,7 +57,7 @@ theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
(h : x ∈ derivedSeries R L 1) : χ x = 0 :=
by
rw [derived_series_def, derived_series_of_ideal_succ, derived_series_of_ideal_zero, ←
- LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
+ LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
apply Submodule.span_induction h
· rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩; apply lie_character_apply_lie
· exact χ.map_zero
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ 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.Abelian
-import Mathbin.Algebra.Lie.Solvable
-import Mathbin.LinearAlgebra.Dual
+import Algebra.Lie.Abelian
+import Algebra.Lie.Solvable
+import LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
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.character
-! 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.Abelian
import Mathbin.Algebra.Lie.Solvable
import Mathbin.LinearAlgebra.Dual
+#align_import algebra.lie.character from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
/-!
# Characters of Lie algebras
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -48,10 +48,12 @@ abbrev LieCharacter :=
variable {R L}
+#print LieAlgebra.lieCharacter_apply_lie /-
@[simp]
theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆ = 0 := by
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
#align lie_algebra.lie_character_apply_lie LieAlgebra.lieCharacter_apply_lie
+-/
#print LieAlgebra.lieCharacter_apply_of_mem_derived /-
theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
@@ -67,6 +69,7 @@ theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
#align lie_algebra.lie_character_apply_of_mem_derived LieAlgebra.lieCharacter_apply_of_mem_derived
-/
+#print LieAlgebra.lieCharacterEquivLinearDual /-
/-- For an Abelian Lie algebra, characters are just linear forms. -/
@[simps]
def lieCharacterEquivLinearDual [IsLieAbelian L] : LieCharacter R L ≃ Module.Dual R L
@@ -80,6 +83,7 @@ def lieCharacterEquivLinearDual [IsLieAbelian L] : LieCharacter R L ≃ Module.D
left_inv χ := by ext; rfl
right_inv ψ := by ext; rfl
#align lie_algebra.lie_character_equiv_linear_dual LieAlgebra.lieCharacterEquivLinearDual
+-/
end LieAlgebra
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.character
-! leanprover-community/mathlib commit 132328c4dd48da87adca5d408ca54f315282b719
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.LinearAlgebra.Dual
/-!
# Characters of Lie algebras
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
A character of a Lie algebra `L` over a commutative ring `R` is a morphism of Lie algebras `L → R`,
where `R` is regarded as a Lie algebra over itself via the ring commutator. For an Abelian Lie
algebra (e.g., a Cartan subalgebra of a semisimple Lie algebra) a character is just a linear form.
mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725
@@ -36,10 +36,12 @@ namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
+#print LieAlgebra.LieCharacter /-
/-- A character of a Lie algebra is a morphism to the scalars. -/
abbrev LieCharacter :=
L →ₗ⁅R⁆ R
#align lie_algebra.lie_character LieAlgebra.LieCharacter
+-/
variable {R L}
@@ -48,6 +50,7 @@ theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
#align lie_algebra.lie_character_apply_lie LieAlgebra.lieCharacter_apply_lie
+#print LieAlgebra.lieCharacter_apply_of_mem_derived /-
theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
(h : x ∈ derivedSeries R L 1) : χ x = 0 :=
by
@@ -59,6 +62,7 @@ theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
· intro y z hy hz; rw [LieHom.map_add, hy, hz, add_zero]
· intro t y hy; rw [LieHom.map_smul, hy, smul_zero]
#align lie_algebra.lie_character_apply_of_mem_derived LieAlgebra.lieCharacter_apply_of_mem_derived
+-/
/-- For an Abelian Lie algebra, characters are just linear forms. -/
@[simps]
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -52,7 +52,7 @@ theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
(h : x ∈ derivedSeries R L 1) : χ x = 0 :=
by
rw [derived_series_def, derived_series_of_ideal_succ, derived_series_of_ideal_zero, ←
- LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
+ LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
apply Submodule.span_induction h
· rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩; apply lie_character_apply_lie
· exact χ.map_zero
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -54,13 +54,10 @@ theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
rw [derived_series_def, derived_series_of_ideal_succ, derived_series_of_ideal_zero, ←
LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
apply Submodule.span_induction h
- · rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩
- apply lie_character_apply_lie
+ · rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩; apply lie_character_apply_lie
· exact χ.map_zero
- · intro y z hy hz
- rw [LieHom.map_add, hy, hz, add_zero]
- · intro t y hy
- rw [LieHom.map_smul, hy, smul_zero]
+ · intro y z hy hz; rw [LieHom.map_add, hy, hz, add_zero]
+ · intro t y hy; rw [LieHom.map_smul, hy, smul_zero]
#align lie_algebra.lie_character_apply_of_mem_derived LieAlgebra.lieCharacter_apply_of_mem_derived
/-- For an Abelian Lie algebra, characters are just linear forms. -/
@@ -73,12 +70,8 @@ def lieCharacterEquivLinearDual [IsLieAbelian L] : LieCharacter R L ≃ Module.D
map_lie' := fun x y => by
rw [LieModule.IsTrivial.trivial, LieRing.of_associative_ring_bracket, mul_comm, sub_self,
LinearMap.toFun_eq_coe, LinearMap.map_zero] }
- left_inv χ := by
- ext
- rfl
- right_inv ψ := by
- ext
- rfl
+ left_inv χ := by ext; rfl
+ right_inv ψ := by ext; rfl
#align lie_algebra.lie_character_equiv_linear_dual LieAlgebra.lieCharacterEquivLinearDual
end LieAlgebra
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -2,16 +2,13 @@
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.character
-! leanprover-community/mathlib commit 132328c4dd48da87adca5d408ca54f315282b719
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
+#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
+
/-!
# Characters of Lie algebras
The unported dependencies are