DibyashanuPati/PhysLean
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor: Lint

Browse source
This commit is contained in:
jstoobysmith 2024-10-15 11:39:40 +00:00
parent 89d1b1a50b
commit f72d69e2ba
6 changed files with 84 additions and 69 deletions
Download patch file Download diff file Expand all files Collapse all files

11
HepLean/SpaceTime/WeylFermion/Contraction.lean
View file

@ -21,7 +21,6 @@ open MatrixGroups
open Complex
open TensorProduct
/-!
## Contraction of Weyl fermions.
@ -76,7 +75,6 @@ def altLeftBi : altLeftHanded →ₗ[ℂ] leftHanded →ₗ[ℂ] ℂ where
simp only [_root_.map_smul, smul_dotProduct, vec2_dotProduct, Fin.isValue, smul_eq_mul,
LinearMap.coe_mk, AddHom.coe_mk, RingHom.id_apply, LinearMap.smul_apply]
/-- The bi-linear map corresponding to contraction of a right-handed Weyl fermion with a
alt-right-handed Weyl fermion. -/
def rightAltBi : rightHanded →ₗ[ℂ] altRightHanded →ₗ[ℂ] ℂ where
@ -170,7 +168,8 @@ The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift rightAltBi
comm M := TensorProduct.ext' fun ψ φ => by
change (M.1.map star *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
change (M.1.map star *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) =
ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
have h1 : (M.1)⁻¹ᴴ = ((M.1)⁻¹.map star)ᵀ := by rfl
rw [dotProduct_mulVec, h1, vecMul_transpose, mulVec_mulVec]
have h2 : ((M.1)⁻¹.map star * (M.1).map star) = 1 := by
@ -193,8 +192,9 @@ def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2
-/
def altRightContraction : altRightHanded ⊗ rightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift altRightBi
comm M := TensorProduct.ext' fun φ ψ => by
change (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1.map star *ᵥ ψ.toFin2ℂ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
comm M := TensorProduct.ext' fun φ ψ => by
change (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1.map star *ᵥ ψ.toFin2ℂ) =
φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
have h1 : (M.1)⁻¹ᴴ = ((M.1)⁻¹.map star)ᵀ := by rfl
rw [dotProduct_mulVec, h1, mulVec_transpose, vecMul_vecMul]
have h2 : ((M.1)⁻¹.map star * (M.1).map star) = 1 := by
@ -251,7 +251,6 @@ informal_lemma altLeftWeylContraction_invariant where
the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
deps :≈ [``altLeftContraction]
informal_lemma rightAltWeylContraction_invariant where
math :≈ "The contraction rightAltWeylContraction is invariant with respect to
the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."

27
HepLean/SpaceTime/WeylFermion/Metric.lean
View file

@ -13,7 +13,7 @@ import HepLean.SpaceTime.WeylFermion.Two
We define the metrics for Weyl fermions, often denoted `ε` in the literature.
These allow us to go from left-handed to alt-left-handed Weyl fermions and back,
and from right-handed to alt-right-handed Weyl fermions and back.
and from right-handed to alt-right-handed Weyl fermions and back.
-/
@ -49,7 +49,7 @@ lemma metricRaw_comm (M : SL(2,ℂ)) : metricRaw * M.1 = (M.1⁻¹)ᵀ * metricR
mul_one, smul_empty, tail_cons, neg_smul, mul_neg, neg_cons, neg_neg, neg_zero, neg_empty,
empty_vecMul, add_cons, empty_add_empty, empty_mul, Equiv.symm_apply_apply]
lemma star_comm_metricRaw (M : SL(2,ℂ)) : M.1.map star * metricRaw = metricRaw * ((M.1)⁻¹)ᴴ := by
lemma star_comm_metricRaw (M : SL(2,ℂ)) : M.1.map star * metricRaw = metricRaw * ((M.1)⁻¹)ᴴ := by
rw [metricRaw]
rw [SpaceTime.SL2C.inverse_coe, eta_fin_two M.1]
rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
@ -83,11 +83,13 @@ def leftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ leftHanded where
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • leftMetricVal =
(TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) (x' • leftMetricVal)
simp
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
apply congrArg
simp [leftMetricVal]
erw [leftLeftToMatrix_ρ_symm]
@ -114,11 +116,13 @@ def altLeftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHande
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • altLeftMetricVal =
(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) (x' • altLeftMetricVal)
simp
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
apply congrArg
simp [altLeftMetricVal]
erw [altLeftaltLeftToMatrix_ρ_symm]
@ -127,7 +131,6 @@ def altLeftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHande
simp only [SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq, one_ne_zero,
not_false_eq_true, mul_nonsing_inv, mul_one]
/-- The metric `ε_{dot a}_{dot a}` as an element of `(rightHanded ⊗ rightHanded).V`. -/
def rightMetricVal : (rightHanded ⊗ rightHanded).V :=
rightRightToMatrix.symm (- metricRaw)
@ -146,7 +149,9 @@ def rightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ rightHanded wher
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • rightMetricVal =
(TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) (x' • rightMetricVal)
@ -186,7 +191,9 @@ def altRightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ altRightHa
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • altRightMetricVal =
(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) (x' • altRightMetricVal)
@ -201,7 +208,7 @@ def altRightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ altRightHa
have h1 : ((M.1).map star * (M.1)⁻¹ᴴᵀ) = 1 := by
refine transpose_eq_one.mp ?_
rw [@transpose_mul]
simp
simp only [transpose_transpose, RCLike.star_def]
change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
rw [← @conjTranspose_mul]
simp

57
HepLean/SpaceTime/WeylFermion/Two.lean
View file

@ -10,7 +10,6 @@ import Mathlib.LinearAlgebra.TensorProduct.Matrix
# Tensor product of two Weyl fermion
-/
namespace Fermion
@ -96,7 +95,7 @@ lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
((leftBasis.tensorProduct leftBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct leftBasis)
(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) v)
(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -108,7 +107,7 @@ lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
erw [Finset.sum_product]
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
have h1 : ∑ x : Fin 2, (∑ j : Fin 2, M.1 i j * leftLeftToMatrix v j x) * M.1 j x
= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 j x := by
= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 j x := by
congr
funext x
rw [Finset.sum_mul]
@ -138,8 +137,8 @@ lemma altLeftaltLeftToMatrix_ρ (v : (altLeftHanded ⊗ altLeftHanded).V) (M : S
((altLeftBasis.tensorProduct altLeftBasis).repr v)))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct altLeftBasis)
(altLeftBasis.tensorProduct altLeftBasis)
(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) v)
(altLeftBasis.tensorProduct altLeftBasis)
(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -178,8 +177,8 @@ lemma leftAltLeftToMatrix_ρ (v : (leftHanded ⊗ altLeftHanded).V) (M : SL(2,
((leftBasis.tensorProduct altLeftBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct altLeftBasis)
(leftBasis.tensorProduct altLeftBasis)
(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) v)
(leftBasis.tensorProduct altLeftBasis)
(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -219,8 +218,8 @@ lemma altLeftLeftToMatrix_ρ (v : (altLeftHanded ⊗ leftHanded).V) (M : SL(2,
((altLeftBasis.tensorProduct leftBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct leftBasis)
(altLeftBasis.tensorProduct leftBasis)
(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) v)
(altLeftBasis.tensorProduct leftBasis)
(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -260,7 +259,8 @@ lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)
((rightBasis.tensorProduct rightBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct rightBasis)
(rightBasis.tensorProduct rightBasis) (TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) v)
(rightBasis.tensorProduct rightBasis)
(TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -271,8 +271,9 @@ lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)
* rightRightToMatrix v k.1 k.2) = _
erw [Finset.sum_product]
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x:= by
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) *
(M.1.map star) j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x:= by
congr
funext x
rw [Finset.sum_mul]
@ -300,8 +301,8 @@ lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M
((altRightBasis.tensorProduct altRightBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct altRightBasis)
(altRightBasis.tensorProduct altRightBasis)
(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) v)
(altRightBasis.tensorProduct altRightBasis)
(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -312,8 +313,9 @@ lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M
* altRightAltRightToMatrix v k.1 k.2) = _
erw [Finset.sum_product]
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) *
(↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
congr
funext x
rw [Finset.sum_mul]
@ -340,8 +342,8 @@ lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(
((rightBasis.tensorProduct altRightBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct altRightBasis)
(rightBasis.tensorProduct altRightBasis)
(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
(rightBasis.tensorProduct altRightBasis)
(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -352,8 +354,9 @@ lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(
* rightAltRightToMatrix v k.1 k.2) = _
erw [Finset.sum_product]
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x)
* (↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
congr
funext x
rw [Finset.sum_mul]
@ -381,8 +384,8 @@ lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(
((altRightBasis.tensorProduct rightBasis).repr (v))))
· apply congrArg
have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct rightBasis)
(altRightBasis.tensorProduct rightBasis)
(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
(altRightBasis.tensorProduct rightBasis)
(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
erw [h1]
rfl
rw [TensorProduct.toMatrix_map]
@ -393,8 +396,10 @@ lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(
* altRightRightToMatrix v k.1 k.2) = _
erw [Finset.sum_product]
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x := by
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2,
(↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) *
(M.1.map star) j x := by
congr
funext x
rw [Finset.sum_mul]
@ -464,7 +469,7 @@ lemma altRightAltRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(
lemma rightAltRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M) (rightAltRightToMatrix.symm v) =
rightAltRightToMatrix.symm ((M.1.map star) * v * (((M.1⁻¹).conjTranspose)ᵀ) ) := by
rightAltRightToMatrix.symm ((M.1.map star) * v * (((M.1⁻¹).conjTranspose)ᵀ)) := by
have h1 := rightAltRightToMatrix_ρ (rightAltRightToMatrix.symm v) M
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
rw [← h1]
@ -478,7 +483,5 @@ lemma altRightRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,
rw [← h1]
simp
end
end Fermion

38
HepLean/SpaceTime/WeylFermion/Unit.lean
View file

@ -42,13 +42,15 @@ def leftAltLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • leftAltLeftUnitVal =
(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) (x' • leftAltLeftUnitVal)
simp
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
apply congrArg
simp [leftAltLeftUnitVal]
simp only [Action.instMonoidalCategory_tensorObj_V, leftAltLeftUnitVal]
erw [leftAltLeftToMatrix_ρ_symm]
apply congrArg
simp
@ -71,11 +73,13 @@ def altLeftLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • altLeftLeftUnitVal =
(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) (x' • altLeftLeftUnitVal)
simp
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
apply congrArg
simp [altLeftLeftUnitVal]
erw [altLeftLeftToMatrix_ρ_symm]
@ -103,20 +107,22 @@ def rightAltRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHa
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • rightAltRightUnitVal =
(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) (x' • rightAltRightUnitVal)
simp
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
apply congrArg
simp [rightAltRightUnitVal]
simp only [Action.instMonoidalCategory_tensorObj_V, rightAltRightUnitVal]
erw [rightAltRightToMatrix_ρ_symm]
apply congrArg
simp
simp only [RCLike.star_def, mul_one]
symm
refine transpose_eq_one.mp ?h.h.h.a
simp
change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
simp only [transpose_mul, transpose_transpose]
change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
rw [@conjTranspose_nonsing_inv]
simp
@ -140,18 +146,20 @@ def altRightRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHa
rfl}
comm M := by
ext x : 2
simp
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
Function.comp_apply]
let x' : ℂ := x
change x' • altRightRightUnitVal =
(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) (x' • altRightRightUnitVal)
simp
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
apply congrArg
simp [altRightRightUnitVal]
erw [altRightRightToMatrix_ρ_symm]
apply congrArg
simp
simp only [mul_one, RCLike.star_def]
symm
change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
rw [@conjTranspose_nonsing_inv]
simp