| import Mathlib |
| import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra |
|
|
| namespace OAI |
|
|
| /-! Calabi Frame Algebra. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus |
| variable {n : ℕ} |
|
|
| def pullConnectionTensor (C : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ConnectionTensor n := |
| fun i => C⁻¹*(∑ p, C p i • T p)*C |
|
|
| lemma tensor_trace_changeBasis (C P M T U : Matrix (Fin n) (Fin n) ℂ) (hC : IsUnit C) : |
| ((C⁻¹*T*C)*(C⁻¹*P*(Cᴴ)⁻¹)*(C⁻¹*U*C)ᴴ*(Cᴴ*M*C)).trace = (T*P*Uᴴ*M).trace := by |
| have hc : C*C⁻¹=1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hC) |
| have hd : (Cᴴ)⁻¹*Cᴴ=1 := by |
| rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one] |
| simp only [Matrix.conjTranspose_mul,Matrix.conjTranspose_nonsing_inv,Matrix.mul_assoc] |
| rw [← Matrix.mul_assoc C C⁻¹,hc,Matrix.one_mul, |
| ← Matrix.mul_assoc (Cᴴ)⁻¹ Cᴴ,hd,Matrix.one_mul, |
| ← Matrix.mul_assoc (Cᴴ)⁻¹ Cᴴ,hd,Matrix.one_mul] |
| calc |
| _ = ((C⁻¹*(T*P*Uᴴ*M))*C).trace := by simp only [mul_assoc] |
| _ = ((C*C⁻¹)*(T*P*Uᴴ*M)).trace := Matrix.trace_mul_cycle _ _ _ |
| _ = _ := by simp only [hc,one_mul,mul_assoc] |
|
|
| lemma tensor_pair_mixed (P M Q C : Matrix (Fin n) (Fin n) ℂ) (T U : ConnectionTensor n) : |
| (∑ i, ∑ j, Q i j * ((∑ p, C p i • T p)*P*(∑ q, C q j • U q)ᴴ*M).trace) = |
| ∑ p, ∑ q, (C*Q*Cᴴ) p q*(T p*P*(U q)ᴴ*M).trace := by |
| have he (i j) : ((∑ p, C p i • T p)*P*(∑ q, C q j • U q)ᴴ*M).trace = |
| ∑ p, ∑ q, C p i * star (C q j) * (T p*P*(U q)ᴴ*M).trace := by |
| simp only [Matrix.sum_mul,Matrix.conjTranspose_sum,Matrix.trace_sum, |
| Matrix.smul_mul,Matrix.mul_smul,Matrix.conjTranspose_smul,Matrix.trace_smul, |
| smul_eq_mul,Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro p _ |
| apply Finset.sum_congr rfl |
| intro q _ |
| ring |
| simp_rw [he] |
| simp only [Matrix.mul_apply,Matrix.conjTranspose_apply,Finset.mul_sum,Finset.sum_mul] |
| calc |
| _ = ∑ p, ∑ q, ∑ i, ∑ j, |
| Q i j * (C p i * star (C q j) * (T p*P*(U q)ᴴ*M).trace) := by |
| conv_lhs => arg 2; ext i; rw [Finset.sum_comm]; arg 2; ext p; rw [Finset.sum_comm] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro p _ |
| rw [Finset.sum_comm] |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro p _ |
| apply Finset.sum_congr rfl |
| intro q _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
|
|
| lemma tensorPairAt_changeBasis (C P M : Matrix (Fin n) (Fin n) ℂ) (hC : IsUnit C) |
| (T U : ConnectionTensor n) : |
| tensorPairAt (C⁻¹*P*(Cᴴ)⁻¹) (Cᴴ*M*C) (pullConnectionTensor C T) (pullConnectionTensor C U) = |
| tensorPairAt P M T U := by |
| unfold tensorPairAt pullConnectionTensor |
| simp_rw [tensor_trace_changeBasis C P M _ _ hC] |
| rw [tensor_pair_mixed] |
| have hc : C*C⁻¹=1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hC) |
| have hd : (Cᴴ)⁻¹*Cᴴ=1 := by |
| rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one] |
| have he : C*(C⁻¹*P*(Cᴴ)⁻¹)*Cᴴ = P := by |
| simp only [mul_assoc] |
| rw [← mul_assoc C C⁻¹,hc,one_mul] |
| simp only [hd,mul_one] |
| simp only [he] |
| end KaehlerCalculus |
|
|
| end |
| end |
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| end OAI |
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|
