515 lines
21 KiB
Text
515 lines
21 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.LinearAlgebra.PiTensorProduct
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import Mathlib.LinearAlgebra.DFinsupp
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/-!
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# Structure of Tensors
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This file sets up the structure `TensorStructure` which contains
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data of types (or `colors`) of indices, the dual of colors, the associated module,
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contraction between color modules, and the unit of contraction.
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This structure is extended to `DualizeTensorStructure` which adds a metric to the tensor structure,
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allowing a vector to be taken to its dual vector by contraction with a specified metric.
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The definition of `DualizeTensorStructure` can be found in
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`HepLean.SpaceTime.LorentzTensor.RisingLowering`.
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The structure `DualizeTensorStructure` is further extended in
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`HepLean.SpaceTime.LorentzTensor.LorentzTensorStruct` to add a group action on the tensor space,
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under which contraction and rising and lowering etc, are invariant.
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## References
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-- For modular operads see: [Raynor][raynor2021graphical]
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-/
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noncomputable section
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open TensorProduct
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variable {R : Type} [CommSemiring R]
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def contrDualLeftAux {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 contrDualMidAux {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] (f : V1 ⊗[R] V2 →ₗ[R] R) :
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(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.assoc R _ _ _).toLinearMap
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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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/-- The allowed colors of indices.
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For example for a real Lorentz tensor these are `{up, down}`. -/
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Color : Type
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/-- To each color we associate a module. -/
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ColorModule : Color → Type
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/-- A map taking every color to its dual color. -/
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τ : Color → Color
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/-- The map `τ` is an involution. -/
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τ_involutive : Function.Involutive τ
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/-- Each `ColorModule` has the structure of an additive commutative monoid. -/
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colorModule_addCommMonoid : ∀ μ, AddCommMonoid (ColorModule μ)
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/-- Each `ColorModule` has the structure of a module over `R`. -/
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colorModule_module : ∀ μ, Module R (ColorModule μ)
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/-- The contraction of a vector with a vector with dual color. -/
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contrDual : ∀ μ, ColorModule μ ⊗[R] ColorModule (τ μ) →ₗ[R] R
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/-- The contraction is symmetric. -/
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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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/-- 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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/-- 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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namespace TensorStructure
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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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instance : AddCommMonoid (𝓣.ColorModule μ) := 𝓣.colorModule_addCommMonoid μ
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instance : Module R (𝓣.ColorModule μ) := 𝓣.colorModule_module μ
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/-- The type of tensors given a map from an indexing set `X` to the type of colors,
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specifying the color of that index. -/
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def Tensor (c : X → 𝓣.Color) : Type := ⨂[R] x, 𝓣.ColorModule (c x)
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instance : AddCommMonoid (𝓣.Tensor cX) :=
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PiTensorProduct.instAddCommMonoid fun i => 𝓣.ColorModule (cX i)
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instance : Module R (𝓣.Tensor cX) := PiTensorProduct.instModule
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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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invFun := (Equiv.cast (congrArg 𝓣.ColorModule h)).symm
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map_add' x y := by
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subst h
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rfl
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map_smul' x y := by
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subst h
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rfl
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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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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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= g (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)) : f = g := by
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apply TensorProduct.ext'
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refine fun x ↦
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PiTensorProduct.induction_on' x ?_ (by
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intro a b hx hy y
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simp [map_add, add_tmul, hx, hy])
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intro rx fx
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refine fun y ↦
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PiTensorProduct.induction_on' y ?_ (by
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intro a b hx hy
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simp at hx hy
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simp [map_add, tmul_add, hx, hy])
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intro ry fy
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simp only [PiTensorProduct.tprodCoeff_eq_smul_tprod, tmul_smul, LinearMapClass.map_smul]
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apply congrArg
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simp only [smul_tmul, tmul_smul, LinearMapClass.map_smul]
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exact congrArg (HSMul.hSMul rx) (h fx fy)
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/-!
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## Mapping isomorphisms
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-/
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/-- An linear equivalence of tensor spaces given a color-preserving equivalence of indexing sets. -/
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def mapIso {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = d ∘ e) :
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𝓣.Tensor c ≃ₗ[R] 𝓣.Tensor d :=
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(PiTensorProduct.reindex R _ e) ≪≫ₗ
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(PiTensorProduct.congr (fun y => 𝓣.colorModuleCast (by rw [h]; simp)))
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lemma mapIso_trans_cond {e : X ≃ Y} {e' : Y ≃ Z} (h : cX = cY ∘ e) (h' : cY = cZ ∘ e') :
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cX = cZ ∘ (e.trans e') := by
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funext a
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subst h h'
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simp
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@[simp]
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lemma mapIso_trans (e : X ≃ Y) (e' : Y ≃ Z) (h : cX = cY ∘ e) (h' : cY = cZ ∘ e') :
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(𝓣.mapIso e h ≪≫ₗ 𝓣.mapIso e' h') = 𝓣.mapIso (e.trans e') (𝓣.mapIso_trans_cond h h') := by
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refine LinearEquiv.toLinearMap_inj.mp ?_
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apply PiTensorProduct.ext
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apply MultilinearMap.ext
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intro x
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simp only [mapIso, LinearMap.compMultilinearMap_apply, LinearEquiv.coe_coe,
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LinearEquiv.trans_apply, PiTensorProduct.reindex_tprod, Equiv.symm_trans_apply]
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change (PiTensorProduct.congr fun y => 𝓣.colorModuleCast _)
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((PiTensorProduct.reindex R (fun x => 𝓣.ColorModule (cY x)) e')
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((PiTensorProduct.congr fun y => 𝓣.colorModuleCast _) _)) =
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(PiTensorProduct.congr fun y => 𝓣.colorModuleCast _)
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((PiTensorProduct.reindex R (fun x => 𝓣.ColorModule (cX x)) (e.trans e')) _)
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rw [PiTensorProduct.congr_tprod, PiTensorProduct.reindex_tprod,
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PiTensorProduct.congr_tprod, PiTensorProduct.reindex_tprod, PiTensorProduct.congr]
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simp [colorModuleCast]
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@[simp]
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lemma mapIso_mapIso (e : X ≃ Y) (e' : Y ≃ Z) (h : cX = cY ∘ e) (h' : cY = cZ ∘ e')
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(T : 𝓣.Tensor cX) :
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(𝓣.mapIso e' h') (𝓣.mapIso e h T) = 𝓣.mapIso (e.trans e') (𝓣.mapIso_trans_cond h h') T := by
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rw [← LinearEquiv.trans_apply, mapIso_trans]
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@[simp]
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lemma mapIso_symm (e : X ≃ Y) (h : cX = cY ∘ e) :
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(𝓣.mapIso e h).symm = 𝓣.mapIso e.symm ((Equiv.eq_comp_symm e cY cX).mpr h.symm) := by
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refine LinearEquiv.toLinearMap_inj.mp ?_
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apply PiTensorProduct.ext
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apply MultilinearMap.ext
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intro x
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simp [mapIso, LinearMap.compMultilinearMap_apply, LinearEquiv.coe_coe,
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LinearEquiv.symm_apply_apply, PiTensorProduct.reindex_tprod]
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change (PiTensorProduct.reindex R (fun x => 𝓣.ColorModule (cX x)) e).symm
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((PiTensorProduct.congr fun y => 𝓣.colorModuleCast _).symm ((PiTensorProduct.tprod R) x)) =
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(PiTensorProduct.congr fun y => 𝓣.colorModuleCast _)
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((PiTensorProduct.reindex R (fun x => 𝓣.ColorModule (cY x)) e.symm)
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((PiTensorProduct.tprod R) x))
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rw [PiTensorProduct.reindex_tprod, PiTensorProduct.congr_tprod, PiTensorProduct.congr_symm_tprod,
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LinearEquiv.symm_apply_eq, PiTensorProduct.reindex_tprod]
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apply congrArg
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funext i
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simp only [colorModuleCast, Equiv.cast_symm, LinearEquiv.coe_symm_mk,
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Equiv.symm_symm_apply, LinearEquiv.coe_mk]
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rw [← Equiv.symm_apply_eq]
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simp only [Equiv.cast_symm, Equiv.cast_apply, cast_cast]
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apply cast_eq_iff_heq.mpr
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rw [Equiv.apply_symm_apply]
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@[simp]
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lemma mapIso_refl : 𝓣.mapIso (Equiv.refl X) (rfl : cX = cX) = LinearEquiv.refl R _ := by
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refine LinearEquiv.toLinearMap_inj.mp ?_
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apply PiTensorProduct.ext
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apply MultilinearMap.ext
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intro x
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simp only [mapIso, Equiv.refl_symm, Equiv.refl_apply, PiTensorProduct.reindex_refl,
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LinearMap.compMultilinearMap_apply, LinearEquiv.coe_coe, LinearEquiv.trans_apply,
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LinearEquiv.refl_apply, LinearEquiv.refl_toLinearMap, LinearMap.id, LinearMap.coe_mk,
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AddHom.coe_mk, id_eq]
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change (PiTensorProduct.congr fun y => 𝓣.colorModuleCast _) ((PiTensorProduct.tprod R) x) = _
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rw [PiTensorProduct.congr_tprod]
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rfl
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@[simp]
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lemma mapIso_tprod {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = d ∘ e)
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(f : (i : X) → 𝓣.ColorModule (c i)) : (𝓣.mapIso e h) (PiTensorProduct.tprod R f) =
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(PiTensorProduct.tprod R (fun i => 𝓣.colorModuleCast (by rw [h]; simp) (f (e.symm i)))) := by
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simp [mapIso]
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change (PiTensorProduct.congr fun y => 𝓣.colorModuleCast _)
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((PiTensorProduct.reindex R _ e) ((PiTensorProduct.tprod R) f)) = _
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rw [PiTensorProduct.reindex_tprod]
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exact PiTensorProduct.congr_tprod (fun y => 𝓣.colorModuleCast _) fun i => f (e.symm i)
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/-!
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## Pure tensors
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This section is needed since: `PiTensorProduct.tmulEquiv` is not defined for dependent types.
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Hence we need to construct a version of it here.
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-/
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/-- The type of pure tensors, i.e. of the form `v1 ⊗ v2 ⊗ v3 ⊗ ...`. -/
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abbrev PureTensor (c : X → 𝓣.Color) := (x : X) → 𝓣.ColorModule (c x)
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/-- A pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` constructed from a pure tensor
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in `𝓣.PureTensor cX` and a pure tensor in `𝓣.PureTensor cY`. -/
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def elimPureTensor (p : 𝓣.PureTensor cX) (q : 𝓣.PureTensor cY) : 𝓣.PureTensor (Sum.elim cX cY) :=
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fun x =>
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match x with
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| Sum.inl x => p x
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| Sum.inr x => q x
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@[simp]
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lemma elimPureTensor_update_right (p : 𝓣.PureTensor cX) (q : 𝓣.PureTensor cY)
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(y : Y) (r : 𝓣.ColorModule (cY y)) : 𝓣.elimPureTensor p (Function.update q y r) =
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Function.update (𝓣.elimPureTensor p q) (Sum.inr y) r := by
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funext x
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match x with
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| Sum.inl x => rfl
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| Sum.inr x =>
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change Function.update q y r x = _
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simp only [Function.update, Sum.inr.injEq, Sum.elim_inr]
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split_ifs
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rename_i h
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subst h
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simp_all only
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rfl
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@[simp]
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lemma elimPureTensor_update_left (p : 𝓣.PureTensor cX) (q : 𝓣.PureTensor cY)
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(x : X) (r : 𝓣.ColorModule (cX x)) : 𝓣.elimPureTensor (Function.update p x r) q =
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Function.update (𝓣.elimPureTensor p q) (Sum.inl x) r := by
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funext y
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match y with
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| Sum.inl y =>
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change (Function.update p x r) y = _
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simp only [Function.update, Sum.inl.injEq, Sum.elim_inl]
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split_ifs
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rename_i h
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subst h
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simp_all only
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rfl
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| Sum.inr y => rfl
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/-- The projection of a pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` onto a pure tensor in
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`𝓣.PureTensor cX`. -/
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def inlPureTensor (p : 𝓣.PureTensor (Sum.elim cX cY)) : 𝓣.PureTensor cX := fun x => p (Sum.inl x)
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/-- The projection of a pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` onto a pure tensor in
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`𝓣.PureTensor cY`. -/
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def inrPureTensor (p : 𝓣.PureTensor (Sum.elim cX cY)) : 𝓣.PureTensor cY := fun y => p (Sum.inr y)
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@[simp]
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lemma inlPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim cX cY)) (x : X)
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(v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inl x))) :
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𝓣.inlPureTensor (Function.update f (Sum.inl x) v1) =
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Function.update (𝓣.inlPureTensor f) x v1 := by
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funext y
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simp [inlPureTensor, Function.update, Sum.inl.injEq, Sum.elim_inl]
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split
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next h =>
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subst h
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simp_all only
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rfl
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@[simp]
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lemma inrPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim cX cY)) (x : X)
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(v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inl x))) :
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𝓣.inrPureTensor (Function.update f (Sum.inl x) v1) = (𝓣.inrPureTensor f) := by
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funext x
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simp [inrPureTensor, Function.update]
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@[simp]
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lemma inrPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim cX cY)) (y : Y)
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(v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inr y))) :
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𝓣.inrPureTensor (Function.update f (Sum.inr y) v1) =
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Function.update (𝓣.inrPureTensor f) y v1 := by
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funext y
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simp [inrPureTensor, Function.update, Sum.inl.injEq, Sum.elim_inl]
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split
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next h =>
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subst h
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simp_all only
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rfl
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@[simp]
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lemma inlPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim cX cY)) (y : Y)
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(v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inr y))) :
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𝓣.inlPureTensor (Function.update f (Sum.inr y) v1) = (𝓣.inlPureTensor f) := by
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funext x
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simp [inlPureTensor, Function.update]
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/-- The multilinear map taking pure tensors a `𝓣.PureTensor cX` and a pure tensor in
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`𝓣.PureTensor cY`, and constructing a tensor in `𝓣.Tensor (Sum.elim cX cY))`. -/
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def elimPureTensorMulLin : MultilinearMap R (fun i => 𝓣.ColorModule (cX i))
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(MultilinearMap R (fun x => 𝓣.ColorModule (cY x)) (𝓣.Tensor (Sum.elim cX cY))) where
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toFun p := {
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toFun := fun q => PiTensorProduct.tprod R (𝓣.elimPureTensor p q)
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map_add' := fun m x v1 v2 => by
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simp [Sum.elim_inl, Sum.elim_inr]
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map_smul' := fun m x r v => by
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simp [Sum.elim_inl, Sum.elim_inr]}
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map_add' p x v1 v2 := by
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apply MultilinearMap.ext
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intro y
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simp
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map_smul' p x r v := by
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apply MultilinearMap.ext
|
||
intro y
|
||
simp
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||
|
||
/-!
|
||
|
||
## tensorator
|
||
|
||
-/
|
||
|
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/-! TODO: Replace with dependent type version of `MultilinearMap.domCoprod` when in Mathlib. -/
|
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/-- The multi-linear map taking a pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` and constructing
|
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a vector in `𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY`. -/
|
||
def domCoprod : MultilinearMap R (fun x => 𝓣.ColorModule (Sum.elim cX cY x))
|
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(𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY) where
|
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toFun f := (PiTensorProduct.tprod R (𝓣.inlPureTensor f)) ⊗ₜ
|
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(PiTensorProduct.tprod R (𝓣.inrPureTensor f))
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map_add' f xy v1 v2:= by
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match xy with
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| Sum.inl x => simp [← TensorProduct.add_tmul]
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| Sum.inr y => simp [← TensorProduct.tmul_add]
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map_smul' f xy r p := by
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||
match xy with
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| Sum.inl x => simp [TensorProduct.tmul_smul, TensorProduct.smul_tmul]
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| Sum.inr y => simp [TensorProduct.tmul_smul, TensorProduct.smul_tmul]
|
||
|
||
/-- The linear map combining two tensors into a single tensor
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||
via the tensor product i.e. `v1 v2 ↦ v1 ⊗ v2`. -/
|
||
def tensoratorSymm : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY →ₗ[R] 𝓣.Tensor (Sum.elim cX cY) := by
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||
refine TensorProduct.lift {
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||
toFun := fun a ↦
|
||
PiTensorProduct.lift <|
|
||
PiTensorProduct.lift (𝓣.elimPureTensorMulLin) a,
|
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map_add' := fun a b ↦ by simp
|
||
map_smul' := fun r a ↦ by simp}
|
||
|
||
/-! TODO: Replace with dependent type version of `PiTensorProduct.tmulEquiv` when in Mathlib. -/
|
||
/-- Splitting a tensor in `𝓣.Tensor (Sum.elim cX cY)` into the tensor product of two tensors. -/
|
||
def tensorator : 𝓣.Tensor (Sum.elim cX cY) →ₗ[R] 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY :=
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PiTensorProduct.lift 𝓣.domCoprod
|
||
|
||
/-- An equivalence formed by taking the tensor product of tensors. -/
|
||
def tensoratorEquiv (c : X → 𝓣.Color) (d : Y → 𝓣.Color) :
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𝓣.Tensor c ⊗[R] 𝓣.Tensor d ≃ₗ[R] 𝓣.Tensor (Sum.elim c d) :=
|
||
LinearEquiv.ofLinear (𝓣.tensoratorSymm) (𝓣.tensorator)
|
||
(by
|
||
apply PiTensorProduct.ext
|
||
apply MultilinearMap.ext
|
||
intro p
|
||
simp [tensorator, tensoratorSymm, domCoprod]
|
||
change (PiTensorProduct.lift _) ((PiTensorProduct.tprod R) _) =
|
||
LinearMap.id ((PiTensorProduct.tprod R) p)
|
||
rw [PiTensorProduct.lift.tprod]
|
||
simp [elimPureTensorMulLin, elimPureTensor]
|
||
change (PiTensorProduct.tprod R) _ = _
|
||
apply congrArg
|
||
funext x
|
||
match x with
|
||
| Sum.inl x => rfl
|
||
| Sum.inr x => rfl)
|
||
(by
|
||
apply tensorProd_piTensorProd_ext
|
||
intro p q
|
||
simp [tensorator, tensoratorSymm]
|
||
change (PiTensorProduct.lift 𝓣.domCoprod)
|
||
((PiTensorProduct.lift (𝓣.elimPureTensorMulLin p)) ((PiTensorProduct.tprod R) q)) =_
|
||
rw [PiTensorProduct.lift.tprod]
|
||
simp [elimPureTensorMulLin]
|
||
rfl)
|
||
|
||
@[simp]
|
||
lemma tensoratorEquiv_tmul_tprod (p : 𝓣.PureTensor cX) (q : 𝓣.PureTensor cY) :
|
||
(𝓣.tensoratorEquiv cX cY) ((PiTensorProduct.tprod R) p ⊗ₜ[R] (PiTensorProduct.tprod R) q) =
|
||
(PiTensorProduct.tprod R) (𝓣.elimPureTensor p q) := by
|
||
simp only [tensoratorEquiv, tensoratorSymm, tensorator, domCoprod, LinearEquiv.ofLinear_apply,
|
||
lift.tmul, LinearMap.coe_mk, AddHom.coe_mk, PiTensorProduct.lift.tprod]
|
||
exact PiTensorProduct.lift.tprod q
|
||
|
||
lemma tensoratorEquiv_mapIso_cond {e : X ≃ Y} {e' : Z ≃ Y} {e'' : W ≃ X}
|
||
(h : cX = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') (h'' : bW = cX ∘ e'') :
|
||
Sum.elim bW cZ = Sum.elim cX cY ∘ ⇑(e''.sumCongr e') := by
|
||
subst h h' h''
|
||
funext x
|
||
match x with
|
||
| Sum.inl x => rfl
|
||
| Sum.inr x => rfl
|
||
|
||
@[simp]
|
||
lemma tensoratorEquiv_mapIso (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
|
||
(h : cX = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') (h'' : bW = cX ∘ e'') :
|
||
(TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ (𝓣.tensoratorEquiv cX cY)
|
||
= (𝓣.tensoratorEquiv bW cZ)
|
||
≪≫ₗ (𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'')) := by
|
||
apply LinearEquiv.toLinearMap_inj.mp
|
||
apply tensorProd_piTensorProd_ext
|
||
intro p q
|
||
simp only [LinearEquiv.coe_coe, LinearEquiv.trans_apply, congr_tmul, mapIso_tprod,
|
||
tensoratorEquiv_tmul_tprod, Equiv.sumCongr_symm, Equiv.sumCongr_apply]
|
||
apply congrArg
|
||
funext x
|
||
match x with
|
||
| Sum.inl x => rfl
|
||
| Sum.inr x => rfl
|
||
|
||
@[simp]
|
||
lemma tensoratorEquiv_mapIso_apply (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
|
||
(h : cX = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') (h'' : cW = cX ∘ e'')
|
||
(x : 𝓣.Tensor cW ⊗[R] 𝓣.Tensor cZ) :
|
||
(𝓣.tensoratorEquiv cX cY) ((TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) x) =
|
||
(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h''))
|
||
((𝓣.tensoratorEquiv cW cZ) x) := by
|
||
trans ((TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ
|
||
(𝓣.tensoratorEquiv cX cY)) x
|
||
rfl
|
||
rw [tensoratorEquiv_mapIso]
|
||
rfl
|
||
exact e
|
||
exact h
|
||
|
||
lemma tensoratorEquiv_mapIso_tmul (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
|
||
(h : cX = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') (h'' : cW = cX ∘ e'')
|
||
(x : 𝓣.Tensor cW) (y : 𝓣.Tensor cZ) :
|
||
(𝓣.tensoratorEquiv cX cY) ((𝓣.mapIso e'' h'' x) ⊗ₜ[R] (𝓣.mapIso e' h' y)) =
|
||
(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h''))
|
||
((𝓣.tensoratorEquiv cW cZ) (x ⊗ₜ y)) := by
|
||
rw [← tensoratorEquiv_mapIso_apply]
|
||
rfl
|
||
exact e
|
||
exact h
|
||
|
||
/-!
|
||
|
||
## Splitting tensors into tensor products
|
||
|
||
-/
|
||
|
||
/-- The decomposition of a set into a direct sum based on the image of an injection. -/
|
||
def decompEmbedSet (f : Y ↪ X) :
|
||
X ≃ {x // x ∈ (Finset.image f Finset.univ)ᶜ} ⊕ Y :=
|
||
(Equiv.Set.sumCompl (Set.range ⇑f)).symm.trans <|
|
||
(Equiv.sumComm _ _).trans <|
|
||
Equiv.sumCongr ((Equiv.subtypeEquivRight (by simp))) <|
|
||
(Function.Embedding.toEquivRange f).symm
|
||
|
||
/-- The restriction of a map from an indexing set to the space to the complement of the image
|
||
of an embedding. -/
|
||
def decompEmbedColorLeft (c : X → 𝓣.Color) (f : Y ↪ X) :
|
||
{x // x ∈ (Finset.image f Finset.univ)ᶜ} → 𝓣.Color :=
|
||
(c ∘ (decompEmbedSet f).symm) ∘ Sum.inl
|
||
|
||
/-- The restriction of a map from an indexing set to the space to the image
|
||
of an embedding. -/
|
||
def decompEmbedColorRight (c : X → 𝓣.Color) (f : Y ↪ X) : Y → 𝓣.Color :=
|
||
(c ∘ (decompEmbedSet f).symm) ∘ Sum.inr
|
||
|
||
lemma decompEmbed_cond (c : X → 𝓣.Color) (f : Y ↪ X) : c =
|
||
(Sum.elim (𝓣.decompEmbedColorLeft c f) (𝓣.decompEmbedColorRight c f)) ∘ decompEmbedSet f := by
|
||
simpa [decompEmbedColorLeft, decompEmbedColorRight] using (Equiv.comp_symm_eq _ _ _).mp rfl
|
||
|
||
/-- Decomposes a tensor into a tensor product of two tensors
|
||
one which has indices in the image of the given embedding, and the other has indices in
|
||
the complement of that image. -/
|
||
def decompEmbed (f : Y ↪ X) :
|
||
𝓣.Tensor cX ≃ₗ[R] 𝓣.Tensor (𝓣.decompEmbedColorLeft cX f) ⊗[R] 𝓣.Tensor (cX ∘ f) :=
|
||
(𝓣.mapIso (decompEmbedSet f) (𝓣.decompEmbed_cond cX f)) ≪≫ₗ
|
||
(𝓣.tensoratorEquiv (𝓣.decompEmbedColorLeft cX f) (𝓣.decompEmbedColorRight cX f)).symm
|
||
|
||
end TensorStructure
|
||
|
||
end
|