feat: Add rising and lower indices

HepLean/SpaceTime/LorentzTensor/Basic.lean

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

HepLean/SpaceTime/LorentzTensor/Contraction.lean

@@ -3,7 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
-import HepLean.SpaceTime.LorentzTensor.Basic
+import HepLean.SpaceTime.LorentzTensor.MulActionTensor
 /-!
 
 # Contraction of indices
@@ -30,9 +30,13 @@ We define a number of ways to contract indices of tensors:
   `𝓣.Tensor (Sum.elim cW cX) ⊗[R] 𝓣.Tensor (Sum.elim cY cZ) →ₗ[R] 𝓣.Tensor (Sum.elim cW cZ)`
 
 -/
+
+/-! TODO: Define contraction based on an equivalence `(C ⊗ C) ⊗ P ≃ X` satisfying ... . -/
+
 noncomputable section
 
 open TensorProduct
+open MulActionTensor
 
 variable {R : Type} [CommSemiring R]
 
@@ -40,11 +44,14 @@ namespace TensorStructure
 
 variable (𝓣 : TensorStructure R)
 
-variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
+variable {d : ℕ} {X Y Y' Z W C P : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
   [Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
+  [Fintype C] [DecidableEq C] [Fintype P] [DecidableEq P]
   {cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
   {cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
 
+variable {G H : Type} [Group G] [Group H] [MulActionTensor G 𝓣]
+local infixl:101 " • " => 𝓣.rep
 /-!
 
 # Contractions of vectors
@@ -55,7 +62,7 @@ variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [
   `𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η` to form a vector in `𝓣.ColorModule η`. -/
 def contrDualLeft {ν η : 𝓣.Color} :
     𝓣.ColorModule ν ⊗[R] 𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η →ₗ[R] 𝓣.ColorModule η :=
-  contrDualLeftAux (𝓣.contrDual ν)
+  contrLeftAux (𝓣.contrDual ν)
 
 /-- The contraction of a vector in `𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule ν` with a vector in
   `𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η` to form a vector in
@@ -63,7 +70,7 @@ def contrDualLeft {ν η : 𝓣.Color} :
 def contrDualMid {μ ν η : 𝓣.Color} :
     (𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule ν) ⊗[R] (𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η) →ₗ[R]
       𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule η :=
-  contrDualMidAux (𝓣.contrDual ν)
+  contrMidAux (𝓣.contrDual ν)
 
 /-- A linear map taking tensors mapped with the same index set to the product of paired tensors. -/
 def pairProd : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cX2 →ₗ[R]
@@ -248,4 +255,104 @@ def contrElim (e : X ≃ Y) (h : cX = 𝓣.τ ∘ cY ∘ e) :
     (TensorProduct.congr (𝓣.tensoratorEquiv cW cX).symm
       (𝓣.tensoratorEquiv cY cZ).symm).toLinearMap
 
+/-!
+
+## Group acting on contraction
+
+-/
+
+@[simp]
+lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) (g : G) :
+    𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.contrAll e h := by
+  apply TensorProduct.ext'
+  refine fun x ↦ PiTensorProduct.induction_on' x ?_ (by
+      intro a b hx hy y
+      simp [map_add, add_tmul, hx, hy])
+  intro rx fx
+  refine fun y ↦ PiTensorProduct.induction_on' y ?_ (by
+      intro a b hx hy
+      simp at hx hy
+      simp [map_add, tmul_add, hx, hy])
+  intro ry fy
+  simp [contrAll, TensorProduct.smul_tmul]
+  apply congrArg
+  apply congrArg
+  simp [contrAll']
+  apply congrArg
+  simp [pairProd]
+  change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
+    (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _)
+  rw [PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map_tprod,
+  PiTensorProduct.map_tprod]
+  simp only [mk_apply]
+  apply congrArg
+  funext x
+  rw [← repColorModule_colorModuleCast_apply]
+  nth_rewrite 2 [← contrDual_inv (c x) g]
+  rfl
+
+@[simp]
+lemma contrAll_rep_apply {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
+    (g : G) (x : 𝓣.Tensor c ⊗ 𝓣.Tensor d) :
+    𝓣.contrAll e h (TensorProduct.map (𝓣.rep g) (𝓣.rep g) x) = 𝓣.contrAll e h x := by
+  change (𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x = _
+  rw [contrAll_rep]
+
+@[simp]
+lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
+    (g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
+    𝓣.contrAll e h ((g • x) ⊗ₜ[R] (g • y)) = 𝓣.contrAll e h (x ⊗ₜ[R] y) := by
+  nth_rewrite 2 [← @contrAll_rep_apply R _ 𝓣 _ _ _ G]
+  rfl
+
+/-!
+
+## Contraction based on specification
+
+-/
+
+lemma contr_cond (e : (C ⊕ C) ⊕ P ≃ X) :
+    cX = Sum.elim (Sum.elim (cX ∘ ⇑e ∘ Sum.inl ∘ Sum.inl) (cX ∘ ⇑e ∘ Sum.inl ∘ Sum.inr)) (
+      cX ∘ ⇑e ∘ Sum.inr) ∘ ⇑e.symm := by
+  rw [Equiv.eq_comp_symm]
+  funext x
+  match x with
+  | Sum.inl (Sum.inl x) => rfl
+  | Sum.inl (Sum.inr x) => rfl
+  | Sum.inr x => rfl
+
+/-- Contraction of indices based on an equivalence `(C ⊕ C) ⊕ P ≃ X`. The indices
+  in `C` are contracted pair-wise, whilst the indices in `P` are preserved. -/
+def contr (e : (C ⊕ C) ⊕ P ≃ X)
+    (h : cX ∘ e ∘ Sum.inl ∘ Sum.inl = 𝓣.τ ∘ cX ∘ e ∘ Sum.inl ∘ Sum.inr) :
+    𝓣.Tensor cX →ₗ[R] 𝓣.Tensor (cX ∘ e ∘ Sum.inr) :=
+  (TensorProduct.lid R _).toLinearMap ∘ₗ
+  (TensorProduct.map (𝓣.contrAll (Equiv.refl C) (by simpa using h)) LinearMap.id) ∘ₗ
+  (TensorProduct.congr (𝓣.tensoratorEquiv _ _).symm (LinearEquiv.refl R _)).toLinearMap ∘ₗ
+  (𝓣.tensoratorEquiv _ _).symm.toLinearMap ∘ₗ
+  (𝓣.mapIso e.symm (𝓣.contr_cond e)).toLinearMap
+
+/-- The contraction of indices via `contr` is equivariant. -/
+@[simp]
+lemma contr_equivariant (e : (C ⊕ C) ⊕ P ≃ X)
+    (h : cX ∘ e ∘ Sum.inl ∘ Sum.inl = 𝓣.τ ∘ cX ∘ e ∘ Sum.inl ∘ Sum.inr)
+    (g : G) (x : 𝓣.Tensor cX) : 𝓣.contr e h (g • x) = g • 𝓣.contr e h x := by
+  simp only [contr, TensorProduct.congr, LinearEquiv.refl_toLinearMap, LinearEquiv.symm_symm,
+    LinearEquiv.refl_symm, LinearEquiv.ofLinear_toLinearMap, LinearEquiv.comp_coe,
+    LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, LinearEquiv.trans_apply,
+    rep_mapIso_apply, rep_tensoratorEquiv_symm_apply]
+  rw [← LinearMap.comp_apply (TensorProduct.map _ _), ← TensorProduct.map_comp]
+  rw [← LinearMap.comp_apply (TensorProduct.map _ _), ← TensorProduct.map_comp]
+  rw [LinearMap.comp_assoc, rep_tensoratorEquiv_symm, ← LinearMap.comp_assoc]
+  simp only [contrAll_rep, LinearMap.comp_id, LinearMap.id_comp]
+  have h1 {M N A B : Type} [AddCommMonoid M] [AddCommMonoid N]
+      [AddCommMonoid A] [AddCommMonoid B] [Module R M] [Module R N] [Module R A] [Module R B]
+      (f : M →ₗ[R] N) (g : A →ₗ[R] B)  : TensorProduct.map f g
+      = TensorProduct.map (LinearMap.id) g ∘ₗ TensorProduct.map f (LinearMap.id) :=
+    ext rfl
+  rw [h1]
+  simp only [LinearMap.coe_comp, Function.comp_apply, rep_lid_apply]
+  rw [← LinearMap.comp_apply (TensorProduct.map _ _), ← TensorProduct.map_comp]
+  rfl
+
 end TensorStructure

HepLean/SpaceTime/LorentzTensor/MulActionTensor.lean

@@ -3,7 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
-import HepLean.SpaceTime.LorentzTensor.Contraction
+import HepLean.SpaceTime.LorentzTensor.Basic
 import Mathlib.RepresentationTheory.Basic
 /-!
 
@@ -124,10 +124,10 @@ lemma rep_mapIso (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) :
 
 @[simp]
 lemma rep_mapIso_apply (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) (x : 𝓣.Tensor cX) :
-    g • (𝓣.mapIso e h x) = (𝓣.mapIso e h) (g • x) := by
+    (𝓣.mapIso e h) (g • x) = g • (𝓣.mapIso e h x)  := by
   trans ((𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap) x
-  rfl
   simp
+  rfl
 
 @[simp]
 lemma rep_tprod (g : G) (f : (i : X) → 𝓣.ColorModule (cX i)) :
@@ -170,54 +170,37 @@ lemma rep_tensoratorEquiv_tmul (g : G) (x : 𝓣.Tensor cX) (y : 𝓣.Tensor cY)
   nth_rewrite 1 [← rep_tensoratorEquiv_apply]
   rfl
 
-/-!
+lemma rep_tensoratorEquiv_symm (g : G) :
-
+    (𝓣.tensoratorEquiv cX cY).symm ∘ₗ 𝓣.rep g  = (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) ∘ₗ
-## Group acting on contraction
+    (𝓣.tensoratorEquiv cX cY).symm.toLinearMap := by
-
+  rw [LinearEquiv.eq_comp_toLinearMap_symm, LinearMap.comp_assoc,
--/
+    LinearEquiv.toLinearMap_symm_comp_eq]
+  exact Eq.symm (rep_tensoratorEquiv 𝓣 g)
 
 @[simp]
-lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) (g : G) :
+lemma rep_tensoratorEquiv_symm_apply (g : G) (x : 𝓣.Tensor (Sum.elim cX cY)) :
-    𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.contrAll e h := by
+    (𝓣.tensoratorEquiv cX cY).symm ((𝓣.rep g) x) =
-  apply TensorProduct.ext'
+    (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) ((𝓣.tensoratorEquiv cX cY).symm x) := by
-  refine fun x ↦ PiTensorProduct.induction_on' x ?_ (by
+  trans ((𝓣.tensoratorEquiv cX cY).symm ∘ₗ 𝓣.rep g) x
-      intro a b hx hy y
+  rfl
-      simp [map_add, add_tmul, hx, hy])
+  rw [rep_tensoratorEquiv_symm]
-  intro rx fx
-  refine fun y ↦ PiTensorProduct.induction_on' y ?_ (by
-      intro a b hx hy
-      simp at hx hy
-      simp [map_add, tmul_add, hx, hy])
-  intro ry fy
-  simp [contrAll, TensorProduct.smul_tmul]
-  apply congrArg
-  apply congrArg
-  simp [contrAll']
-  apply congrArg
-  simp [pairProd]
-  change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
-    (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _)
-  rw [PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map_tprod,
-  PiTensorProduct.map_tprod]
-  simp only [mk_apply]
-  apply congrArg
-  funext x
-  rw [← repColorModule_colorModuleCast_apply]
-  nth_rewrite 2 [← contrDual_inv (c x) g]
   rfl
 
 @[simp]
-lemma contrAll_rep_apply {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
+lemma rep_lid  (g : G)  : TensorProduct.lid R (𝓣.Tensor cX) ∘ₗ
-    (g : G) (x : 𝓣.Tensor c ⊗ 𝓣.Tensor d) :
+    (TensorProduct.map (LinearMap.id) (𝓣.rep g)) = (𝓣.rep g) ∘ₗ
-    𝓣.contrAll e h (TensorProduct.map (𝓣.rep g) (𝓣.rep g) x) = 𝓣.contrAll e h x := by
+    (TensorProduct.lid R (𝓣.Tensor cX)).toLinearMap  := by
-  change (𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x = _
+  apply TensorProduct.ext'
-  rw [contrAll_rep]
+  intro r y
+  simp
 
 @[simp]
-lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
+lemma rep_lid_apply (g : G) (x : R ⊗[R] 𝓣.Tensor cX) :
-    (g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
+    (TensorProduct.lid R (𝓣.Tensor cX)) ((TensorProduct.map (LinearMap.id) (𝓣.rep g)) x) =
-    𝓣.contrAll e h ((g • x) ⊗ₜ[R] (g • y)) = 𝓣.contrAll e h (x ⊗ₜ[R] y) := by
+    (𝓣.rep g) ((TensorProduct.lid R (𝓣.Tensor cX)).toLinearMap x) := by
-  nth_rewrite 2 [← @contrAll_rep_apply R _ G]
+  trans ((TensorProduct.lid R (𝓣.Tensor cX)) ∘ₗ (TensorProduct.map (LinearMap.id) (𝓣.rep g))) x
+  rfl
+  rw [rep_lid]
   rfl
 
 end TensorStructure

HepLean/SpaceTime/LorentzTensor/RisingLowering.lean

@@ -0,0 +1,171 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzTensor.Basic
+import HepLean.SpaceTime.LorentzTensor.MulActionTensor
+/-!
+
+# Rising and Lowering of indices
+
+We use the term `dualize` to describe the more general version of rising and lowering of indices.
+
+In particular, rising and lowering indices corresponds taking the color of that index
+to its dual.
+
+-/
+
+noncomputable section
+
+open TensorProduct
+
+variable {R : Type} [CommSemiring R]
+
+namespace TensorStructure
+
+variable (𝓣 : TensorStructure R)
+
+variable {d : ℕ} {X Y Y' Z W C P : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
+  [Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
+  [Fintype C] [DecidableEq C] [Fintype P] [DecidableEq P]
+  {cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
+  {cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
+
+variable {G H : Type} [Group G] [Group H] [MulActionTensor G 𝓣]
+local infixl:101 " • " => 𝓣.rep
+open MulActionTensor
+
+/-!
+
+## Properties of the unit
+
+-/
+
+/-! TODO: Move -/
+
+lemma unit_lhs_eq (x : 𝓣.ColorModule μ) (y : 𝓣.ColorModule (𝓣.τ μ) ⊗[R] 𝓣.ColorModule μ) :
+    contrLeftAux (𝓣.contrDual μ) (x ⊗ₜ[R] y)  =
+    (contrRightAux (𝓣.contrDual (𝓣.τ μ))) ((TensorProduct.comm R _ _) y
+    ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) x) := by
+  refine  TensorProduct.induction_on y (by simp) ?_ (fun z1 z2 h1 h2 => ?_)
+  intro x1 x2
+  simp only [contrRightAux, LinearEquiv.refl_toLinearMap, comm_tmul, colorModuleCast,
+    Equiv.cast_symm, LinearEquiv.coe_mk, Equiv.cast_apply, LinearMap.coe_comp, LinearEquiv.coe_coe,
+    Function.comp_apply, assoc_tmul, map_tmul, LinearMap.id_coe, id_eq, contrDual_symm', cast_cast,
+    cast_eq, rid_tmul]
+  rfl
+  simp [LinearMap.map_add, add_tmul]
+  rw [← h1, ← h2, tmul_add, LinearMap.map_add]
+
+
+@[simp]
+lemma unit_lid : (contrRightAux (𝓣.contrDual (𝓣.τ μ))) ((TensorProduct.comm R _ _) (𝓣.unit μ)
+    ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) x)  = x := by
+  have h1 := 𝓣.unit_rid μ x
+  rw [← unit_lhs_eq]
+  exact h1
+
+/-!
+
+## Properties of the metric
+
+-/
+
+@[simp]
+lemma metric_cast (h : μ = ν) :
+    (TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast h)) (𝓣.metric μ) =
+    𝓣.metric ν := by
+  subst h
+  erw [congr_refl_refl]
+  simp only [LinearEquiv.refl_apply]
+
+
+
+@[simp]
+lemma metric_contrRight_unit (μ : 𝓣.Color) (x : 𝓣.ColorModule μ ) :
+    (contrRightAux (𝓣.contrDual μ)) (𝓣.metric μ  ⊗ₜ[R]
+    ((contrRightAux (𝓣.contrDual (𝓣.τ μ)))
+      (𝓣.metric (𝓣.τ μ) ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm x)))) = x := by
+  change (contrRightAux (𝓣.contrDual μ) ∘ₗ TensorProduct.map (LinearMap.id)
+      (contrRightAux (𝓣.contrDual (𝓣.τ μ)))) (𝓣.metric μ
+      ⊗ₜ[R] 𝓣.metric (𝓣.τ μ) ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm x)) = _
+  rw [contrRightAux_comp]
+  simp
+  rw [𝓣.metric_dual]
+  simp only [unit_lid]
+
+/-!
+
+## Dualizing
+
+-/
+
+def dualizeSymm  (μ : 𝓣.Color) : 𝓣.ColorModule (𝓣.τ μ) →ₗ[R] 𝓣.ColorModule μ :=
+  contrRightAux (𝓣.contrDual μ)  ∘ₗ
+    TensorProduct.lTensorHomToHomLTensor R _ _ _ (𝓣.metric μ ⊗ₜ[R] LinearMap.id)
+
+def dualizeFun  (μ : 𝓣.Color) : 𝓣.ColorModule μ →ₗ[R] 𝓣.ColorModule (𝓣.τ μ) :=
+  𝓣.dualizeSymm (𝓣.τ μ) ∘ₗ (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm).toLinearMap
+
+def dualizeModule (μ : 𝓣.Color) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModule (𝓣.τ μ) := by
+  refine LinearEquiv.ofLinear (𝓣.dualizeFun μ) (𝓣.dualizeSymm μ) ?_ ?_
+  · apply LinearMap.ext
+    intro x
+    simp [dualizeFun, dualizeSymm, LinearMap.coe_comp, LinearEquiv.coe_coe,
+      Function.comp_apply, lTensorHomToHomLTensor_apply, LinearMap.id_coe, id_eq,
+      contrDual_symm_contrRightAux_apply_tmul, metric_cast]
+  · apply LinearMap.ext
+    intro x
+    simp only [dualizeSymm, dualizeFun, LinearMap.coe_comp, LinearEquiv.coe_coe,
+      Function.comp_apply, lTensorHomToHomLTensor_apply, LinearMap.id_coe, id_eq,
+      metric_contrRight_unit]
+
+def dualizeAll : 𝓣.Tensor cX ≃ₗ[R] 𝓣.Tensor (𝓣.τ ∘ cX) := by
+  refine LinearEquiv.ofLinear
+    (PiTensorProduct.map (fun x => (𝓣.dualizeModule (cX x)).toLinearMap))
+    (PiTensorProduct.map (fun x => (𝓣.dualizeModule (cX x)).symm.toLinearMap)) ?_ ?_
+  all_goals
+    apply LinearMap.ext
+    refine fun x ↦ PiTensorProduct.induction_on' x ?_ (by
+      intro a b hx a
+      simp [map_add, add_tmul, hx]
+      simp_all only [Function.comp_apply, LinearMap.coe_comp, LinearMap.id_coe, id_eq])
+    intro rx fx
+    simp
+    apply congrArg
+    change (PiTensorProduct.map _)
+      ((PiTensorProduct.map _) ((PiTensorProduct.tprod R) fx)) = _
+    rw [PiTensorProduct.map_tprod, PiTensorProduct.map_tprod]
+    apply congrArg
+    simp
+
+lemma dualize_cond (e : C ⊕ P ≃ X) :
+    cX = Sum.elim (cX ∘ e ∘ Sum.inl) (cX ∘ e ∘ Sum.inr) ∘ e.symm := by
+  rw [Equiv.eq_comp_symm]
+  funext x
+  match x with
+  | Sum.inl x => rfl
+  | Sum.inr x => rfl
+
+lemma dualize_cond' (e : C ⊕ P ≃ X) :
+    Sum.elim (𝓣.τ ∘ cX ∘ ⇑e ∘ Sum.inl) (cX ∘ ⇑e ∘ Sum.inr) =
+    (Sum.elim (𝓣.τ ∘ cX ∘ ⇑e ∘ Sum.inl) (cX ∘ ⇑e ∘ Sum.inr) ∘ ⇑e.symm) ∘ ⇑e := by
+  funext x
+  match x with
+  | Sum.inl x => simp
+  | Sum.inr x => simp
+
+/-! TODO: Show that `dualize` is equivariant with respect to the group action. -/
+
+def dualize (e : C ⊕ P ≃ X) : 𝓣.Tensor cX ≃ₗ[R]
+    𝓣.Tensor (Sum.elim (𝓣.τ ∘ cX ∘ e ∘ Sum.inl) (cX ∘ e ∘ Sum.inr) ∘ e.symm) :=
+  𝓣.mapIso e.symm (𝓣.dualize_cond e) ≪≫ₗ
+  (𝓣.tensoratorEquiv _ _).symm ≪≫ₗ
+  TensorProduct.congr 𝓣.dualizeAll (LinearEquiv.refl _ _) ≪≫ₗ
+  (𝓣.tensoratorEquiv _ _) ≪≫ₗ
+  𝓣.mapIso e (𝓣.dualize_cond' e)
+
+end TensorStructure
+
+end