refactor: Lint
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jstoobysmith
parent
89d1b1a50b
commit
f72d69e2ba
6 changed files with 84 additions and 69 deletions
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@ -21,7 +21,6 @@ open MatrixGroups
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open Complex
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open TensorProduct
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/-!
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## Contraction of Weyl fermions.
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@ -76,7 +75,6 @@ def altLeftBi : altLeftHanded →ₗ[ℂ] leftHanded →ₗ[ℂ] ℂ where
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simp only [_root_.map_smul, smul_dotProduct, vec2_dotProduct, Fin.isValue, smul_eq_mul,
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LinearMap.coe_mk, AddHom.coe_mk, RingHom.id_apply, LinearMap.smul_apply]
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/-- The bi-linear map corresponding to contraction of a right-handed Weyl fermion with a
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alt-right-handed Weyl fermion. -/
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def rightAltBi : rightHanded →ₗ[ℂ] altRightHanded →ₗ[ℂ] ℂ where
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@ -170,7 +168,8 @@ The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
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def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
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hom := TensorProduct.lift rightAltBi
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comm M := TensorProduct.ext' fun ψ φ => by
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change (M.1.map star *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
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change (M.1.map star *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) =
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ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
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have h1 : (M.1)⁻¹ᴴ = ((M.1)⁻¹.map star)ᵀ := by rfl
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rw [dotProduct_mulVec, h1, vecMul_transpose, mulVec_mulVec]
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have h2 : ((M.1)⁻¹.map star * (M.1).map star) = 1 := by
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@ -193,8 +192,9 @@ def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2
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-/
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def altRightContraction : altRightHanded ⊗ rightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
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hom := TensorProduct.lift altRightBi
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comm M := TensorProduct.ext' fun φ ψ => by
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change (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1.map star *ᵥ ψ.toFin2ℂ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
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comm M := TensorProduct.ext' fun φ ψ => by
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change (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1.map star *ᵥ ψ.toFin2ℂ) =
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φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
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have h1 : (M.1)⁻¹ᴴ = ((M.1)⁻¹.map star)ᵀ := by rfl
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rw [dotProduct_mulVec, h1, mulVec_transpose, vecMul_vecMul]
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have h2 : ((M.1)⁻¹.map star * (M.1).map star) = 1 := by
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@ -251,7 +251,6 @@ informal_lemma altLeftWeylContraction_invariant where
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the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
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deps :≈ [``altLeftContraction]
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informal_lemma rightAltWeylContraction_invariant where
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math :≈ "The contraction rightAltWeylContraction is invariant with respect to
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the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
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@ -13,7 +13,7 @@ import HepLean.SpaceTime.WeylFermion.Two
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We define the metrics for Weyl fermions, often denoted `ε` in the literature.
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These allow us to go from left-handed to alt-left-handed Weyl fermions and back,
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and from right-handed to alt-right-handed Weyl fermions and back.
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and from right-handed to alt-right-handed Weyl fermions and back.
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-/
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@ -49,7 +49,7 @@ lemma metricRaw_comm (M : SL(2,ℂ)) : metricRaw * M.1 = (M.1⁻¹)ᵀ * metricR
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mul_one, smul_empty, tail_cons, neg_smul, mul_neg, neg_cons, neg_neg, neg_zero, neg_empty,
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empty_vecMul, add_cons, empty_add_empty, empty_mul, Equiv.symm_apply_apply]
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lemma star_comm_metricRaw (M : SL(2,ℂ)) : M.1.map star * metricRaw = metricRaw * ((M.1)⁻¹)ᴴ := by
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lemma star_comm_metricRaw (M : SL(2,ℂ)) : M.1.map star * metricRaw = metricRaw * ((M.1)⁻¹)ᴴ := by
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rw [metricRaw]
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rw [SpaceTime.SL2C.inverse_coe, eta_fin_two M.1]
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rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
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@ -83,11 +83,13 @@ def leftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ leftHanded where
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rfl}
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comm M := by
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ext x : 2
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simp
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
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Function.comp_apply]
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let x' : ℂ := x
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change x' • leftMetricVal =
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(TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) (x' • leftMetricVal)
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simp
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simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
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apply congrArg
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simp [leftMetricVal]
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erw [leftLeftToMatrix_ρ_symm]
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@ -114,11 +116,13 @@ def altLeftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHande
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rfl}
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comm M := by
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ext x : 2
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simp
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
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Function.comp_apply]
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let x' : ℂ := x
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change x' • altLeftMetricVal =
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) (x' • altLeftMetricVal)
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simp
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simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
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apply congrArg
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simp [altLeftMetricVal]
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erw [altLeftaltLeftToMatrix_ρ_symm]
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@ -127,7 +131,6 @@ def altLeftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHande
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simp only [SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq, one_ne_zero,
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not_false_eq_true, mul_nonsing_inv, mul_one]
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/-- The metric `ε_{dot a}_{dot a}` as an element of `(rightHanded ⊗ rightHanded).V`. -/
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def rightMetricVal : (rightHanded ⊗ rightHanded).V :=
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rightRightToMatrix.symm (- metricRaw)
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@ -146,7 +149,9 @@ def rightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ rightHanded wher
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rfl}
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comm M := by
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ext x : 2
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simp
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
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Function.comp_apply]
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let x' : ℂ := x
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change x' • rightMetricVal =
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(TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) (x' • rightMetricVal)
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@ -186,7 +191,9 @@ def altRightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ altRightHa
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rfl}
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comm M := by
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ext x : 2
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simp
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
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Function.comp_apply]
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let x' : ℂ := x
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change x' • altRightMetricVal =
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) (x' • altRightMetricVal)
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@ -201,7 +208,7 @@ def altRightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ altRightHa
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have h1 : ((M.1).map star * (M.1)⁻¹ᴴᵀ) = 1 := by
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refine transpose_eq_one.mp ?_
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rw [@transpose_mul]
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simp
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simp only [transpose_transpose, RCLike.star_def]
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change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
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rw [← @conjTranspose_mul]
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simp
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@ -10,7 +10,6 @@ import Mathlib.LinearAlgebra.TensorProduct.Matrix
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# Tensor product of two Weyl fermion
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-/
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namespace Fermion
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@ -96,7 +95,7 @@ lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
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((leftBasis.tensorProduct leftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct leftBasis)
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(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) v)
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(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -108,7 +107,7 @@ lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ j : Fin 2, M.1 i j * leftLeftToMatrix v j x) * M.1 j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 j x := by
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= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -138,8 +137,8 @@ lemma altLeftaltLeftToMatrix_ρ (v : (altLeftHanded ⊗ altLeftHanded).V) (M : S
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((altLeftBasis.tensorProduct altLeftBasis).repr v)))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct altLeftBasis)
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(altLeftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) v)
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(altLeftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -178,8 +177,8 @@ lemma leftAltLeftToMatrix_ρ (v : (leftHanded ⊗ altLeftHanded).V) (M : SL(2,
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((leftBasis.tensorProduct altLeftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct altLeftBasis)
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(leftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) v)
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(leftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -219,8 +218,8 @@ lemma altLeftLeftToMatrix_ρ (v : (altLeftHanded ⊗ leftHanded).V) (M : SL(2,
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((altLeftBasis.tensorProduct leftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct leftBasis)
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(altLeftBasis.tensorProduct leftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) v)
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(altLeftBasis.tensorProduct leftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -260,7 +259,8 @@ lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)
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((rightBasis.tensorProduct rightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct rightBasis)
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(rightBasis.tensorProduct rightBasis) (TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) v)
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(rightBasis.tensorProduct rightBasis)
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(TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -271,8 +271,9 @@ lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)
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* rightRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x:= by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) *
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(M.1.map star) j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
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((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x:= by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -300,8 +301,8 @@ lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M
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((altRightBasis.tensorProduct altRightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct altRightBasis)
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(altRightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) v)
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(altRightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -312,8 +313,9 @@ lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M
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* altRightAltRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) *
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(↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
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((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -340,8 +342,8 @@ lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(
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((rightBasis.tensorProduct altRightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct altRightBasis)
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(rightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
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(rightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -352,8 +354,9 @@ lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(
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* rightAltRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x)
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* (↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
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((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -381,8 +384,8 @@ lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(
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((altRightBasis.tensorProduct rightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct rightBasis)
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(altRightBasis.tensorProduct rightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
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(altRightBasis.tensorProduct rightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -393,8 +396,10 @@ lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(
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* altRightRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x := by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2,
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(↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) *
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(M.1.map star) j x := by
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congr
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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
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
||||
|
|
|
|||
|
|
@ -46,14 +46,14 @@ def discreteFunctorMapIso {c1 c2 : Discrete C} (h : c1 ⟶ c2) :
|
|||
|
||||
lemma discreteFun_hom_trans {c1 c2 c3 : Discrete C} (h1 : c1 = c2) (h2 : c2 = c3)
|
||||
(v : F.obj c1) : (F.map (eqToHom h2)).hom ((F.map (eqToHom h1)).hom v)
|
||||
= (F.map (eqToHom ( h1.trans h2))).hom v := by
|
||||
= (F.map (eqToHom (h1.trans h2))).hom v := by
|
||||
subst h2 h1
|
||||
simp_all only [eqToHom_refl, Discrete.functor_map_id, Action.id_hom, ModuleCat.id_apply]
|
||||
|
||||
/-- The linear equivalence between `(F.obj c1).V ≃ₗ[k] (F.obj c2).V` induced by an equality of
|
||||
`c1` and `c2`. -/
|
||||
def discreteFunctorMapEqIso {c1 c2 : Discrete C} (h : c1.as = c2.as) :
|
||||
(F.obj c1).V ≃ₗ[k] (F.obj c2).V := LinearEquiv.ofLinear
|
||||
(F.obj c1).V ≃ₗ[k] (F.obj c2).V := LinearEquiv.ofLinear
|
||||
(F.mapIso (Discrete.eqToIso h)).hom.hom (F.mapIso (Discrete.eqToIso h)).inv.hom
|
||||
(by
|
||||
ext x : 2
|
||||
|
|
@ -63,7 +63,7 @@ def discreteFunctorMapEqIso {c1 c2 : Discrete C} (h : c1.as = c2.as) :
|
|||
rw [ModuleCat.ext_iff] at h1
|
||||
have h1x := h1 x
|
||||
simp only [CategoryStruct.comp] at h1x
|
||||
simpa using h1x )
|
||||
simpa using h1x)
|
||||
(by
|
||||
ext x : 2
|
||||
simp only [Functor.mapIso_inv, eqToIso.inv, Functor.mapIso_hom, eqToIso.hom, LinearMap.coe_comp,
|
||||
|
|
@ -122,7 +122,6 @@ lemma objObj'_ρ_empty (g : G) : (objObj' F (𝟙_ (OverColor C))).ρ g = Linear
|
|||
funext i
|
||||
exact Empty.elim i
|
||||
|
||||
|
||||
open TensorProduct in
|
||||
@[simp]
|
||||
lemma objObj'_V_carrier (f : OverColor C) :
|
||||
|
|
@ -149,7 +148,6 @@ lemma mapToLinearEquiv'_tprod {f g : OverColor C} (m : f ⟶ g)
|
|||
rw [PiTensorProduct.reindex_tprod, PiTensorProduct.congr_tprod]
|
||||
rfl
|
||||
|
||||
|
||||
/-- Given a morphism in `OverColor C` the corresopnding map of representations induced by
|
||||
reindexing. -/
|
||||
def objMap' {f g : OverColor C} (m : f ⟶ g) : objObj' F f ⟶ objObj' F g where
|
||||
|
|
@ -231,7 +229,6 @@ lemma μModEquiv_tmul_tprod {X Y : OverColor C}
|
|||
rw [PiTensorProduct.congr_tprod]
|
||||
rfl
|
||||
|
||||
|
||||
/-- The natural isomorphism corresponding to the tensorate. -/
|
||||
def μ (X Y : OverColor C) : objObj' F X ⊗ objObj' F Y ≅ objObj' F (X ⊗ Y) :=
|
||||
Action.mkIso (μModEquiv F X Y).toModuleIso
|
||||
|
|
@ -263,7 +260,6 @@ lemma μ_tmul_tprod {X Y : OverColor C} (p : (i : X.left) → F.obj (Discrete.mk
|
|||
discreteSumEquiv F i (HepLean.PiTensorProduct.elimPureTensor p q i) := by
|
||||
exact μModEquiv_tmul_tprod F p q
|
||||
|
||||
|
||||
lemma μ_natural_left {X Y : OverColor C} (f : X ⟶ Y) (Z : OverColor C) :
|
||||
MonoidalCategory.whiskerRight (objMap' F f) (objObj' F Z) ≫ (μ F Y Z).hom =
|
||||
(μ F X Z).hom ≫ objMap' F (MonoidalCategory.whiskerRight f Z) := by
|
||||
|
|
@ -301,7 +297,6 @@ lemma μ_natural_left {X Y : OverColor C} (f : X ⟶ Y) (Z : OverColor C) :
|
|||
Functor.mapIso_refl, Iso.refl_hom, Action.id_hom, Iso.refl_inv]
|
||||
rfl
|
||||
|
||||
|
||||
lemma μ_natural_right {X Y : OverColor C} (X' : OverColor C) (f : X ⟶ Y) :
|
||||
MonoidalCategory.whiskerLeft (objObj' F X') (objMap' F f) ≫ (μ F X' Y).hom =
|
||||
(μ F X' X).hom ≫ objMap' F (MonoidalCategory.whiskerLeft X' f) := by
|
||||
|
|
@ -337,7 +332,6 @@ lemma μ_natural_right {X Y : OverColor C} (X' : OverColor C) (f : X ⟶ Y) :
|
|||
rfl
|
||||
| Sum.inr i => rfl
|
||||
|
||||
|
||||
lemma associativity (X Y Z : OverColor C) :
|
||||
whiskerRight (μ F X Y).hom (objObj' F Z) ≫
|
||||
(μ F (X ⊗ Y) Z).hom ≫ objMap' F (associator X Y Z).hom =
|
||||
|
|
@ -533,7 +527,7 @@ def mapApp' (X : OverColor C) : (obj' F).obj X ⟶ (obj' F').obj X where
|
|||
funext i
|
||||
simpa using LinearMap.congr_fun ((η.app (Discrete.mk (X.hom i))).comm M) (x i)
|
||||
|
||||
lemma mapApp'_tprod (X : OverColor C) (p : (i : X.left) → F.obj (Discrete.mk (X.hom i))) :
|
||||
lemma mapApp'_tprod (X : OverColor C) (p : (i : X.left) → F.obj (Discrete.mk (X.hom i))) :
|
||||
(mapApp' η X).hom (PiTensorProduct.tprod k p) =
|
||||
PiTensorProduct.tprod k fun i => (η.app (Discrete.mk (X.hom i))).hom (p i) := by
|
||||
change (mapApp' η X).hom (PiTensorProduct.tprod k p) = _
|
||||
|
|
@ -618,7 +612,7 @@ end lift
|
|||
noncomputable def lift : (Discrete C ⥤ Rep k G) ⥤ MonoidalFunctor (OverColor C) (Rep k G) where
|
||||
obj F := lift.obj' F
|
||||
map η := lift.map' η
|
||||
map_id F := by
|
||||
map_id F := by
|
||||
simp only [lift.map']
|
||||
refine MonoidalNatTrans.ext' (fun X => ?_)
|
||||
ext x : 2
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue