/-
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.LorentzVector.Complex.Two
/-!

# Unit for complex Lorentz vectors

-/
noncomputable section

open Matrix
open MatrixGroups
open Complex
open TensorProduct
open SpaceTime
open CategoryTheory.MonoidalCategory
namespace Lorentz

/-- The contra-co unit for complex lorentz vectors. Usually denoted `δⁱᵢ`. -/
def contrCoUnitVal : (complexContr ⊗ complexCo).V :=
  contrCoToMatrix.symm 1

/-- The contra-co unit for complex lorentz vectors as a morphism
  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo`, manifesting the invaraince under
  the `SL(2, ℂ)` action. -/
def contrCoUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo where
  hom := {
    toFun := fun a =>
      let a' : ℂ := a
      a' • contrCoUnitVal,
    map_add' := fun x y => by
      simp only [add_smul],
    map_smul' := fun m x => by
      simp only [smul_smul]
      rfl}
  comm M := by
    ext x : 2
    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' • contrCoUnitVal =
      (TensorProduct.map (complexContr.ρ M) (complexCo.ρ M)) (x' • contrCoUnitVal)
    simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
    apply congrArg
    simp only [Action.instMonoidalCategory_tensorObj_V, contrCoUnitVal]
    erw [contrCoToMatrix_ρ_symm]
    apply congrArg
    simp

/-- The co-contra unit for complex lorentz vectors. Usually denoted `δᵢⁱ`. -/
def coContrUnitVal : (complexCo ⊗ complexContr).V :=
  coContrToMatrix.symm 1

/-- The co-contra unit for complex lorentz vectors as a morphism
  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr`, manifesting the invaraince under
  the `SL(2, ℂ)` action. -/
def coContrUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr where
  hom := {
    toFun := fun a =>
      let a' : ℂ := a
      a' • coContrUnitVal,
    map_add' := fun x y => by
      simp only [add_smul],
    map_smul' := fun m x => by
      simp only [smul_smul]
      rfl}
  comm M := by
    ext x : 2
    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' • coContrUnitVal =
      (TensorProduct.map (complexCo.ρ M) (complexContr.ρ M)) (x' • coContrUnitVal)
    simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
    apply congrArg
    simp only [Action.instMonoidalCategory_tensorObj_V, coContrUnitVal]
    erw [coContrToMatrix_ρ_symm]
    apply congrArg
    symm
    refine transpose_eq_one.mp ?h.h.h.a
    simp

end Lorentz
end