feat: Add rising and lower indices
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jstoobysmith
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4 changed files with 450 additions and 60 deletions
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@ -34,22 +34,58 @@ open TensorProduct
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variable {R : Type} [CommSemiring R]
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namespace TensorStructure
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/-- An auxillary function to contract the vector space `V1` and `V2` in `V1 ⊗[R] V2 ⊗[R] V3`. -/
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def contrDualLeftAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
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def contrLeftAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
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[Module R V1] [Module R V2] [Module R V3] (f : V1 ⊗[R] V2 →ₗ[R] R) :
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V1 ⊗[R] V2 ⊗[R] V3 →ₗ[R] V3 :=
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(TensorProduct.lid R _).toLinearMap ∘ₗ
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TensorProduct.map (f) (LinearEquiv.refl R V3).toLinearMap
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∘ₗ (TensorProduct.assoc R _ _ _).symm.toLinearMap
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def contrRightAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
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[Module R V1] [Module R V2] [Module R V3] (f : V1 ⊗[R] V2 →ₗ[R] R) :
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(V3 ⊗[R] V1) ⊗[R] V2 →ₗ[R] V3 :=
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(TensorProduct.rid R _).toLinearMap ∘ₗ
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TensorProduct.map (LinearEquiv.refl R V3).toLinearMap f ∘ₗ
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(TensorProduct.assoc R _ _ _).toLinearMap
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/-- An auxillary function to contract the vector space `V1` and `V2` in
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`V4 ⊗[R] V1 ⊗[R] V2 ⊗[R] V3`. -/
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def contrDualMidAux {V1 V2 V3 V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
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def contrMidAux {V1 V2 V3 V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
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[AddCommMonoid V4] [Module R V1] [Module R V2] [Module R V3] [Module R V4]
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(f : V1 ⊗[R] V2 →ₗ[R] R) : (V4 ⊗[R] V1) ⊗[R] (V2 ⊗[R] V3) →ₗ[R] V4 ⊗[R] V3 :=
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(TensorProduct.map (LinearEquiv.refl R V4).toLinearMap (contrDualLeftAux f)) ∘ₗ
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(TensorProduct.map (LinearEquiv.refl R V4).toLinearMap (contrLeftAux f)) ∘ₗ
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(TensorProduct.assoc R _ _ _).toLinearMap
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lemma contrRightAux_comp {V1 V2 V3 V4 V5 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
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[AddCommMonoid V4] [AddCommMonoid V5] [Module R V1] [Module R V2] [Module R V3] [Module R V2] [Module R V4]
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[Module R V5]
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(f : V2 ⊗[R] V3 →ₗ[R] R) (g : V4 ⊗[R] V5 →ₗ[R] R) :
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(contrRightAux f ∘ₗ TensorProduct.map (LinearMap.id : V1 ⊗[R] V2 →ₗ[R] V1 ⊗[R] V2)
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(contrRightAux g)) =
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(contrRightAux g) ∘ₗ TensorProduct.map (contrMidAux f) LinearMap.id
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∘ₗ (TensorProduct.assoc R _ _ _).symm.toLinearMap := by
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apply TensorProduct.ext'
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intro x y
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refine TensorProduct.induction_on x (by simp) ?_ (fun x z h1 h2 =>
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by simp [add_tmul, LinearMap.map_add, h1, h2])
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intro x1 x2
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refine TensorProduct.induction_on y (by simp) ?_ (fun x z h1 h2 =>
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by simp [add_tmul, tmul_add, LinearMap.map_add, h1, h2])
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intro y x5
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refine TensorProduct.induction_on y (by simp) ?_ (fun x z h1 h2 =>
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by simp [add_tmul, tmul_add, LinearMap.map_add, h1, h2])
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intro x3 x4
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simp [contrRightAux, contrMidAux, contrLeftAux]
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erw [TensorProduct.map_tmul]
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simp only [LinearMapClass.map_smul, LinearMap.id_coe, id_eq, mk_apply, rid_tmul]
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end TensorStructure
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/-- An initial structure specifying a tensor system (e.g. a system in which you can
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define real Lorentz tensors or Einstein notation convention). -/
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structure TensorStructure (R : Type) [CommSemiring R] where
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@ -72,18 +108,14 @@ structure TensorStructure (R : Type) [CommSemiring R] where
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contrDual_symm : ∀ μ x y, (contrDual μ) (x ⊗ₜ[R] y) =
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(contrDual (τ μ)) (y ⊗ₜ[R] (Equiv.cast (congrArg ColorModule (τ_involutive μ).symm) x))
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/-- The unit of the contraction. -/
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unit : (μ : Color) → ColorModule μ ⊗[R] ColorModule (τ μ)
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unit : (μ : Color) → ColorModule (τ μ) ⊗[R] ColorModule μ
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/-- The unit is a right identity. -/
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unit_lid : ∀ μ (x : ColorModule μ),
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TensorProduct.rid R _
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(TensorProduct.map (LinearEquiv.refl R (ColorModule μ)).toLinearMap
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(contrDual μ ∘ₗ (TensorProduct.comm R _ _).toLinearMap)
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((TensorProduct.assoc R _ _ _) (unit μ ⊗ₜ[R] x))) = x
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unit_rid : ∀ μ (x : ColorModule μ), TensorStructure.contrLeftAux (contrDual μ) (x ⊗ₜ[R] unit μ) = x
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/-- The metric for a given color. -/
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metric : (μ : Color) → ColorModule μ ⊗[R] ColorModule μ
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/-- The metric contracted with its dual is the unit. -/
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metric_dual : ∀ (μ : Color), (contrDualMidAux (contrDual μ)
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(metric μ ⊗ₜ[R] metric (τ μ))) = unit μ
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metric_dual : ∀ (μ : Color), (TensorStructure.contrMidAux (contrDual μ)
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(metric μ ⊗ₜ[R] metric (τ μ))) = TensorProduct.comm _ _ _ (unit μ)
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namespace TensorStructure
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@ -92,7 +124,7 @@ variable (𝓣 : TensorStructure R)
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variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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{cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
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{cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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{cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν η : 𝓣.Color}
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instance : AddCommMonoid (𝓣.ColorModule μ) := 𝓣.colorModule_addCommMonoid μ
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@ -107,6 +139,16 @@ instance : AddCommMonoid (𝓣.Tensor cX) :=
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instance : Module R (𝓣.Tensor cX) := PiTensorProduct.instModule
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/-!
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## Color
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Recall the `color` of an index describes the type of the index.
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For example, in a real Lorentz tensor the colors are `{up, down}`.
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-/
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/-- Equivalence of `ColorModule` given an equality of colors. -/
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def colorModuleCast (h : μ = ν) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModule ν where
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toFun := Equiv.cast (congrArg 𝓣.ColorModule h)
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@ -120,6 +162,45 @@ def colorModuleCast (h : μ = ν) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModu
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left_inv x := Equiv.symm_apply_apply (Equiv.cast (congrArg 𝓣.ColorModule h)) x
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right_inv x := Equiv.apply_symm_apply (Equiv.cast (congrArg 𝓣.ColorModule h)) x
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/-- A relation on colors which is true if the two colors are equal or are duals. -/
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def colorRel (μ ν : 𝓣.Color) : Prop := μ = ν ∨ μ = 𝓣.τ ν
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/-- An equivalence relation on colors which is true if the two colors are equal or are duals. -/
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def colorEquivRel : Equivalence 𝓣.colorRel where
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refl := by
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intro x
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left
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rfl
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symm := by
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intro x y h
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rcases h with h | h
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· left
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exact h.symm
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· right
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subst h
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exact (𝓣.τ_involutive y).symm
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trans := by
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intro x y z hxy hyz
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rcases hxy with hxy | hxy <;>
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rcases hyz with hyz | hyz <;>
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subst hxy hyz
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· left
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rfl
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· right
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rfl
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· right
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rfl
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· left
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exact 𝓣.τ_involutive z
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/-- The structure of a setoid on colors, two colors are related if they are equal,
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or dual. -/
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instance colorSetoid : Setoid 𝓣.Color := ⟨𝓣.colorRel, 𝓣.colorEquivRel⟩
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/-- A map taking a color to its equivalence class in `colorSetoid`. -/
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def colorQuot (μ : 𝓣.Color) : Quotient 𝓣.colorSetoid :=
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Quotient.mk 𝓣.colorSetoid μ
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lemma tensorProd_piTensorProd_ext {M : Type} [AddCommMonoid M] [Module R M]
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{f g : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY →ₗ[R] M}
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(h : ∀ p q, f (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)
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@ -479,6 +560,54 @@ lemma tensoratorEquiv_mapIso_tmul (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
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/-!
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## contrDual properties
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-/
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lemma contrDual_cast (h : μ = ν) (x : 𝓣.ColorModule μ) (y : 𝓣.ColorModule (𝓣.τ μ)) :
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𝓣.contrDual μ (x ⊗ₜ[R] y) = 𝓣.contrDual ν (𝓣.colorModuleCast h x ⊗ₜ[R]
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𝓣.colorModuleCast (congrArg 𝓣.τ h) y) := by
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subst h
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rfl
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/-- `𝓣.contrDual (𝓣.τ μ)` in terms of `𝓣.contrDual μ`. -/
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@[simp]
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lemma contrDual_symm' (μ : 𝓣.Color) (x : 𝓣.ColorModule (𝓣.τ μ))
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(y : 𝓣.ColorModule (𝓣.τ (𝓣.τ μ))) : 𝓣.contrDual (𝓣.τ μ) (x ⊗ₜ[R] y) =
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(𝓣.contrDual μ) ((𝓣.colorModuleCast (𝓣.τ_involutive μ) y) ⊗ₜ[R] x) := by
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rw [𝓣.contrDual_symm, 𝓣.contrDual_cast (𝓣.τ_involutive μ)]
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congr
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simp [colorModuleCast]
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lemma contrDual_symm_contrRightAux (h : ν = η):
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(𝓣.colorModuleCast h) ∘ₗ contrRightAux (𝓣.contrDual μ) =
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contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ))) ∘ₗ
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(TensorProduct.congr (TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm))
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(𝓣.colorModuleCast ((𝓣.τ_involutive (𝓣.τ μ)).symm))).toLinearMap := by
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apply TensorProduct.ext'
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intro x y
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refine TensorProduct.induction_on x (by simp) ?_ ?_
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· intro x z
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simp [contrRightAux]
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congr
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simp [colorModuleCast]
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simp [colorModuleCast]
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· intro x z h1 h2
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simp [add_tmul, LinearMap.map_add, h1, h2]
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lemma contrDual_symm_contrRightAux_apply_tmul (h : ν = η)
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(m : 𝓣.ColorModule ν ⊗[R] 𝓣.ColorModule μ) (x : 𝓣.ColorModule (𝓣.τ μ)) :
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𝓣.colorModuleCast h (contrRightAux (𝓣.contrDual μ) (m ⊗ₜ[R] x)) =
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contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ)))
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((TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) (m)) ⊗ₜ
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(𝓣.colorModuleCast (𝓣.τ_involutive (𝓣.τ μ)).symm x)) := by
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trans ((𝓣.colorModuleCast h) ∘ₗ contrRightAux (𝓣.contrDual μ)) (m ⊗ₜ[R] x)
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rfl
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rw [contrDual_symm_contrRightAux]
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rfl
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/-!
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## Splitting tensors into tensor products
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-/
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