Research proof index

Contraction on the right with contrCoUnit does nothing.

The metric ηᵢᵢ as a morphism 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo, making its invariance under the action of SL(2,ℂ). -/ def coMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap (CoℂModule.SL2CRep.tprod CoℂModule.SL2CRep) where toFun := fun a => let a' : ℂ := a a' • coMetricVal map_add' := fun x y => by simp only [add_smul] map_smu...

The metric tensor is non-degenerate. -/ lemma nondegenerate : (∀ (x : ContrMod d), ⟪x, y⟫ₘ = 0) ↔ y = 0 := by refine Iff.intro (fun h => ?) (fun h => ?) · exact (self_parity_eq_zero_iff _).mp ((symm _ _).trans $ h _) · simp [h] set_option backward.isDefEq.respectTransparency false in lemma matrix_apply_eq_iff_sub : ⟪x, Λ *ᵥ y⟫ₘ = ⟪x, Λ' *ᵥ y⟫ₘ ↔ ⟪x, (Λ...

The metric tensor is non-degenerate. -/ lemma nondegenerate : (∀ (x : ContrMod d), ⟪x, y⟫ₘ = 0) ↔ y = 0 := by refine Iff.intro (fun h => ?) (fun h => ?) · exact (self_parity_eq_zero_iff _).mp ((symm _ _).trans $ h _) · simp [h] set_option backward.isDefEq.respectTransparency false in lemma matrix_apply_eq_iff_sub : ⟪x, Λ *ᵥ y⟫ₘ = ⟪x, Λ' *ᵥ y⟫ₘ ↔ ⟪x, (Λ...

Expanding contrContrToMatrix in terms of the standard basis.

Expanding contrContrToMatrixRe in terms of the standard basis.