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feat: Add symm of unit for complex lorentz

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jstoobysmith 2024-10-24 16:04:05 +00:00
parent 942ee12e60
commit 8c584431c4
3 changed files with 66 additions and 0 deletions
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35
HepLean/SpaceTime/WeylFermion/Unit.lean
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@ -325,5 +325,40 @@ lemma contr_rightAltRightUnit (x : altRightHanded) :
erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul]
simp only [Fin.isValue, one_smul]
/-!
## Symmetry properties of the units
-/
open CategoryTheory
lemma altLeftLeftUnit_symm :
(altLeftLeftUnit.hom (1 : ℂ)) = (altLeftHanded ◁ 𝟙 _).hom ((β_ leftHanded altLeftHanded ).hom.hom
(leftAltLeftUnit.hom (1 : ℂ))) := by
rw [altLeftLeftUnit_apply_one, altLeftLeftUnitVal_expand_tmul]
rw [leftAltLeftUnit_apply_one, leftAltLeftUnitVal_expand_tmul]
rfl
lemma leftAltLeftUnit_symm :
(leftAltLeftUnit.hom (1 : ℂ)) = (leftHanded ◁ 𝟙 _).hom ((β_ altLeftHanded leftHanded ).hom.hom
(altLeftLeftUnit.hom (1 : ℂ))) := by
rw [altLeftLeftUnit_apply_one, altLeftLeftUnitVal_expand_tmul]
rw [leftAltLeftUnit_apply_one, leftAltLeftUnitVal_expand_tmul]
rfl
lemma altRightRightUnit_symm :
(altRightRightUnit.hom (1 : ℂ)) = (altRightHanded ◁ 𝟙 _).hom ((β_ rightHanded altRightHanded ).hom.hom
(rightAltRightUnit.hom (1 : ℂ))) := by
rw [altRightRightUnit_apply_one, altRightRightUnitVal_expand_tmul]
rw [rightAltRightUnit_apply_one, rightAltRightUnitVal_expand_tmul]
rfl
lemma rightAltRightUnit_symm :
(rightAltRightUnit.hom (1 : ℂ)) = (rightHanded ◁ 𝟙 _).hom ((β_ altRightHanded rightHanded ).hom.hom
(altRightRightUnit.hom (1 : ℂ))) := by
rw [altRightRightUnit_apply_one, altRightRightUnitVal_expand_tmul]
rw [rightAltRightUnit_apply_one, rightAltRightUnitVal_expand_tmul]
rfl
end
end Fermion