File size: 3,518 Bytes
2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
end OAI
|