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feat: Add reps to real modules

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jstoobysmith 2024-11-08 06:41:33 +00:00
parent 1350ab732d
commit 6b6f9261ca
4 changed files with 38 additions and 34 deletions
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42
HepLean/SpaceTime/LorentzVector/Real/Modules.lean
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@ -68,18 +68,24 @@ lemma val_smul (r : ℝ) (ψ : ContrℝModule d) : (r • ψ).val = r • ψ.val
/-- The linear equivalence between `ContrℝModule` and `(Fin 1 ⊕ Fin d → ℝ)`. -/
@[simps!]
def toFin1dℝEquiv : ContrℝModule d ≃ₗ[ℝ] (Fin 1 ⊕ Fin d → ℝ) where
toFun := toFin1dℝFun
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl
invFun := toFin1dℝFun.symm
left_inv := fun _ => rfl
right_inv := fun _ => rfl
def toFin1dℝEquiv : ContrℝModule d ≃ₗ[ℝ] (Fin 1 ⊕ Fin d → ℝ) :=
Equiv.linearEquiv ℝ toFin1dℝFun
/-- The underlying element of `Fin 1 ⊕ Fin d → ℝ` of a element in `ContrℝModule` defined
through the linear equivalence `toFin1dℝEquiv`. -/
abbrev toFin1dℝ (ψ : ContrℝModule d) := toFin1dℝEquiv ψ
/-- The standard basis of `ContrℝModule` indexed by `Fin 1 ⊕ Fin d`. -/
@[simps!]
def stdBasis : Basis (Fin 1 ⊕ Fin d) ℝ (ContrℝModule d) := Basis.ofEquivFun toFin1dℝEquiv
/-- The representation of the Lorentz group acting on `ContrℝModule d`. -/
def rep : Representation ℝ (LorentzGroup d) (ContrℝModule d) where
toFun g := Matrix.toLinAlgEquiv stdBasis g
map_one' := (MulEquivClass.map_eq_one_iff (Matrix.toLinAlgEquiv stdBasis)).mpr rfl
map_mul' x y := by
simp only [lorentzGroupIsGroup_mul_coe, _root_.map_mul]
end ContrℝModule
/-- The module for covariant (up-index) complex Lorentz vectors. -/
@ -112,18 +118,26 @@ instance : Module ℝ (CoℝModule d) := Equiv.module ℝ toFin1dℝFun
/-- The linear equivalence between `CoℝModule` and `(Fin 1 ⊕ Fin d → ℝ)`. -/
@[simps!]
def toFin1dℝEquiv : CoℝModule d ≃ₗ[ℝ] (Fin 1 ⊕ Fin d → ℝ) where
toFun := toFin1dℝFun
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl
invFun := toFin1dℝFun.symm
left_inv := fun _ => rfl
right_inv := fun _ => rfl
def toFin1dℝEquiv : CoℝModule d ≃ₗ[ℝ] (Fin 1 ⊕ Fin d → ℝ) :=
Equiv.linearEquiv ℝ toFin1dℝFun
/-- The underlying element of `Fin 1 ⊕ Fin d → ℝ` of a element in `CoℝModule` defined
through the linear equivalence `toFin1dℝEquiv`. -/
abbrev toFin1dℝ (ψ : CoℝModule d) := toFin1dℝEquiv ψ
/-- The standard basis of `CoℝModule` indexed by `Fin 1 ⊕ Fin d`. -/
@[simps!]
def stdBasis : Basis (Fin 1 ⊕ Fin d) ℝ (CoℝModule d) := Basis.ofEquivFun toFin1dℝEquiv
/-- The representation of the Lorentz group acting on `CoℝModule d`. -/
def rep : Representation ℝ (LorentzGroup d) (CoℝModule d) where
toFun g := Matrix.toLinAlgEquiv stdBasis (LorentzGroup.transpose g⁻¹)
map_one' := by
simp only [inv_one, LorentzGroup.transpose_one, lorentzGroupIsGroup_one_coe, _root_.map_one]
map_mul' x y := by
simp only [_root_.mul_inv_rev, lorentzGroupIsGroup_inv, LorentzGroup.transpose_mul,
lorentzGroupIsGroup_mul_coe, _root_.map_mul]
end CoℝModule
end