refactor: Update operator map
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jstoobysmith
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da595e8ad2
commit
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1 changed files with 67 additions and 76 deletions
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@ -16,7 +16,9 @@ noncomputable section
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open FieldStatistic
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/-- A map from the free algebra of fields `FreeAlgebra ℂ I` to an algebra `A`, to be
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variable {𝓕 : Type}
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/-- A map from the free algebra of fields `FreeAlgebra ℂ 𝓕` to an algebra `A`, to be
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thought of as the operator algebra is said to be an operator map if
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all super commutors of fields land in the center of `A`,
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if two fields are of a different grade then their super commutor lands on zero,
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@ -25,22 +27,22 @@ open FieldStatistic
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This can be thought as as a condtion on the operator algebra `A` as much as it can
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on `F`. -/
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class OperatorMap {A : Type} [Semiring A] [Algebra ℂ A]
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(q : I → FieldStatistic) (le1 : I → I → Prop)
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[DecidableRel le1] (F : FreeAlgebra ℂ I →ₐ[ℂ] A) : Prop where
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(q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop)
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[DecidableRel le] (F : FreeAlgebra ℂ 𝓕 →ₐ[ℂ] A) : Prop where
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superCommute_mem_center : ∀ i j, F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) ∈
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Subalgebra.center ℂ A
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superCommute_diff_grade_zero : ∀ i j, q i ≠ q j →
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F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) = 0
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superCommute_ordered_zero : ∀ i j, ∀ a b,
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F (koszulOrder q le1 (a * superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j) * b)) = 0
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F (koszulOrder q le (a * superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j) * b)) = 0
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namespace OperatorMap
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variable {I: Type} {A : Type} [Semiring A] [Algebra ℂ A]
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{q : I → FieldStatistic} {le1 : I → I → Prop}
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[DecidableRel le1] (F : FreeAlgebra ℂ I →ₐ[ℂ] A)
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variable {A : Type} [Semiring A] [Algebra ℂ A]
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{q : 𝓕 → FieldStatistic} {le : 𝓕 → 𝓕 → Prop}
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[DecidableRel le] (F : FreeAlgebra ℂ 𝓕 →ₐ[ℂ] A)
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lemma superCommute_ofList_singleton_ι_center [OperatorMap q le1 F] (i j :I) :
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lemma superCommute_ofList_singleton_ι_center [OperatorMap q le F] (i j : 𝓕) :
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F (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A := by
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have h1 : F (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) =
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xa • F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) := by
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@ -51,22 +53,22 @@ lemma superCommute_ofList_singleton_ι_center [OperatorMap q le1 F] (i j :I) :
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rfl
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rw [h1]
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refine Subalgebra.smul_mem (Subalgebra.center ℂ A) ?_ xa
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exact superCommute_mem_center (le1 := le1) i j
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exact superCommute_mem_center (le := le) i j
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end OperatorMap
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lemma superCommuteSplit_operatorMap {I : Type} (q : I → FieldStatistic)
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(le1 : I → I → Prop) [DecidableRel le1]
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(lb : List I) (xa xb : ℂ) (n : ℕ)
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(hn : n < lb.length) {A : Type} [Semiring A] [Algebra ℂ A] (f : FreeAlgebra ℂ I →ₐ[ℂ] A)
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[OperatorMap q le1 f] (i : I) :
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variable {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le]
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lemma superCommuteSplit_operatorMap (lb : List 𝓕) (xa xb : ℂ) (n : ℕ)
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(hn : n < lb.length) {A : Type} [Semiring A] [Algebra ℂ A] (f : FreeAlgebra ℂ 𝓕 →ₐ[ℂ] A)
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[OperatorMap q le f] (i : 𝓕) :
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f (superCommuteSplit q [i] lb xa xb n hn) =
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f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ (lb.get ⟨n, hn⟩)))
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* (superCommuteCoef q [i] (List.take n lb) •
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f (ofList (List.eraseIdx lb n) xb)) := by
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have hn : f ((superCommute q) (ofList [i] xa) (FreeAlgebra.ι ℂ (lb.get ⟨n, hn⟩))) ∈
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Subalgebra.center ℂ A :=
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OperatorMap.superCommute_ofList_singleton_ι_center (le1 := le1) f i (lb.get ⟨n, hn⟩)
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OperatorMap.superCommute_ofList_singleton_ι_center (le := le) f i (lb.get ⟨n, hn⟩)
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rw [Subalgebra.mem_center_iff] at hn
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rw [superCommuteSplit, map_mul, map_mul, map_smul, hn, mul_assoc, smul_mul_assoc,
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← map_mul, ← ofList_pair]
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@ -74,12 +76,12 @@ lemma superCommuteSplit_operatorMap {I : Type} (q : I → FieldStatistic)
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· exact Eq.symm (List.eraseIdx_eq_take_drop_succ lb n)
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· exact one_mul xb
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lemma superCommuteLiftSplit_operatorMap {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → FieldStatistic) (c : (Σ i, f i)) (r : List I) (x y : ℂ) (n : ℕ)
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lemma superCommuteLiftSplit_operatorMap {f : 𝓕 → Type} [∀ i, Fintype (f i)]
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(c : (Σ i, f i)) (r : List 𝓕) (x y : ℂ) (n : ℕ)
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(hn : n < r.length)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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(le : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le]
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{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
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[OperatorMap (fun i => q i.1) le1 F] :
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[OperatorMap (fun i => q i.1) le F] :
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F (superCommuteLiftSplit q [c] r x y n hn) = superCommuteLiftCoef q [c] (List.take n r) •
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(F (superCommute (fun i => q i.1) (ofList [c] x)
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(sumFiber f (FreeAlgebra.ι ℂ (r.get ⟨n, hn⟩))))
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@ -94,7 +96,7 @@ lemma superCommuteLiftSplit_operatorMap {I : Type} {f : I → Type} [∀ i, Fint
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rw [map_sum, map_sum]
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refine Subalgebra.sum_mem _ ?_
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intro n
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exact fun a => OperatorMap.superCommute_ofList_singleton_ι_center (le1 := le1) F c _
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exact fun a => OperatorMap.superCommute_ofList_singleton_ι_center (le := le) F c _
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rw [Subalgebra.mem_center_iff] at h1
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rw [h1, mul_assoc, ← map_mul]
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congr
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@ -104,30 +106,27 @@ lemma superCommuteLiftSplit_operatorMap {I : Type} {f : I → Type} [∀ i, Fint
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congr
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exact Eq.symm (List.eraseIdx_eq_take_drop_succ r n)
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lemma superCommute_koszulOrder_le_ofList {I : Type}
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(q : I → FieldStatistic) (r : List I) (x : ℂ)
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(le1 :I → I → Prop) [DecidableRel le1] [IsTotal I le1] [IsTrans I le1]
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(i : I)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [OperatorMap q le1 F] :
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F ((superCommute q (FreeAlgebra.ι ℂ i) (koszulOrder q le1 (ofList r x)))) =
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lemma superCommute_koszulOrder_le_ofList [IsTotal 𝓕 le] [IsTrans 𝓕 le] (r : List 𝓕) (x : ℂ)
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(i : 𝓕) {A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ 𝓕 →ₐ A) [OperatorMap q le F] :
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F ((superCommute q (FreeAlgebra.ι ℂ i) (koszulOrder q le (ofList r x)))) =
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∑ n : Fin r.length, (superCommuteCoef q [r.get n] (r.take n)) •
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(F (((superCommute q) (ofList [i] 1)) (FreeAlgebra.ι ℂ (r.get n))) *
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F ((koszulOrder q le1) (ofList (r.eraseIdx ↑n) x))) := by
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F ((koszulOrder q le) (ofList (r.eraseIdx ↑n) x))) := by
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rw [koszulOrder_ofList, map_smul, map_smul, ← ofList_singleton, superCommute_ofList_sum]
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rw [map_sum, ← (HepLean.List.insertionSortEquiv le1 r).sum_comp]
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rw [map_sum, ← (HepLean.List.insertionSortEquiv le r).sum_comp]
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conv_lhs =>
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enter [2, 2]
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intro n
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rw [superCommuteSplit_operatorMap (le1 := le1)]
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rw [superCommuteSplit_operatorMap (le := le)]
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enter [1, 2, 2, 2]
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change ((List.insertionSort le1 r).get ∘ (HepLean.List.insertionSortEquiv le1 r)) n
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change ((List.insertionSort le r).get ∘ (HepLean.List.insertionSortEquiv le r)) n
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rw [HepLean.List.insertionSort_get_comp_insertionSortEquiv]
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conv_lhs =>
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enter [2, 2]
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intro n
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rw [HepLean.List.eraseIdx_insertionSort_fin le1 r n]
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rw [ofList_insertionSort_eq_koszulOrder q le1]
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rw [HepLean.List.eraseIdx_insertionSort_fin le r n]
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rw [ofList_insertionSort_eq_koszulOrder q le]
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rw [Finset.smul_sum]
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conv_lhs =>
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rhs
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@ -136,7 +135,7 @@ lemma superCommute_koszulOrder_le_ofList {I : Type}
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congr
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funext n
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by_cases hq : q i ≠ q (r.get n)
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· have hn := OperatorMap.superCommute_diff_grade_zero (q := q) (F := F) le1 i (r.get n) hq
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· have hn := OperatorMap.superCommute_diff_grade_zero (q := q) (F := F) le i (r.get n) hq
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conv_lhs =>
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enter [2, 1]
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rw [ofList_singleton, hn]
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@ -145,18 +144,16 @@ lemma superCommute_koszulOrder_le_ofList {I : Type}
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rw [ofList_singleton, hn]
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simp
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· congr 1
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trans staticWickCoef q le1 r i n
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trans staticWickCoef q le r i n
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· rw [staticWickCoef, mul_assoc]
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refine staticWickCoef_eq_get q le1 r i n ?_
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refine staticWickCoef_eq_get q le r i n ?_
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simpa using hq
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lemma koszulOrder_of_le_all_ofList {I : Type}
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(q : I → FieldStatistic) (r : List I) (x : ℂ) (le1 : I → I → Prop) [DecidableRel le1]
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(i : I)
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lemma koszulOrder_of_le_all_ofList (r : List 𝓕) (x : ℂ) (i : 𝓕)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [OperatorMap q le1 F] :
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F (koszulOrder q le1 (ofList r x * FreeAlgebra.ι ℂ i))
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= superCommuteCoef q [i] r • F (koszulOrder q le1 (FreeAlgebra.ι ℂ i * ofList r x)) := by
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(F : FreeAlgebra ℂ 𝓕 →ₐ A) [OperatorMap q le F] :
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F (koszulOrder q le (ofList r x * FreeAlgebra.ι ℂ i))
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= superCommuteCoef q [i] r • F (koszulOrder q le (FreeAlgebra.ι ℂ i * ofList r x)) := by
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conv_lhs =>
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enter [2, 2]
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rw [← ofList_singleton]
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@ -184,14 +181,13 @@ lemma koszulOrder_of_le_all_ofList {I : Type}
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simp only [smul_zero, Finset.sum_const_zero, add_zero]
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rw [ofList_singleton]
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lemma le_all_mul_koszulOrder_ofList {I : Type}
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(q : I → FieldStatistic) (r : List I) (x : ℂ) (le1 : I → I→ Prop) [DecidableRel le1]
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(i : I) (hi : ∀ (j : I), le1 j i)
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lemma le_all_mul_koszulOrder_ofList (r : List 𝓕) (x : ℂ)
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(i : 𝓕) (hi : ∀ (j : 𝓕), le j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [OperatorMap q le1 F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder q le1 (ofList r x)) =
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F ((koszulOrder q le1) (FreeAlgebra.ι ℂ i * ofList r x)) +
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F (((superCommute q) (ofList [i] 1)) ((koszulOrder q le1) (ofList r x))) := by
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(F : FreeAlgebra ℂ 𝓕 →ₐ A) [OperatorMap q le F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder q le (ofList r x)) =
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F ((koszulOrder q le) (FreeAlgebra.ι ℂ i * ofList r x)) +
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F (((superCommute q) (ofList [i] 1)) ((koszulOrder q le) (ofList r x))) := by
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rw [koszulOrder_ofList, Algebra.mul_smul_comm, map_smul, ← ofList_singleton,
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ofList_ofList_superCommute q, map_add, smul_add, ← map_smul]
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conv_lhs =>
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@ -201,20 +197,19 @@ lemma le_all_mul_koszulOrder_ofList {I : Type}
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rw [koszulOrder_mul_ge, map_smul]
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congr
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· rw [koszulOrder_of_le_all_ofList]
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rw [superCommuteCoef_perm_snd q [i] (List.insertionSort le1 r) r
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(List.perm_insertionSort le1 r)]
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rw [superCommuteCoef_perm_snd q [i] (List.insertionSort le r) r
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(List.perm_insertionSort le r)]
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rw [smul_smul]
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rw [superCommuteCoef_mul_self]
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simp [ofList_singleton]
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· rw [map_smul, map_smul]
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· exact fun j => hi j
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/-- In the expansions of `F (FreeAlgebra.ι ℂ i * koszulOrder le1 q (ofList r x))`
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the ter multiplying `F ((koszulOrder le1 q) (ofList (optionEraseZ r i n) x))`. -/
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def superCommuteCenterOrder {I : Type}
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(q : I → FieldStatistic) (r : List I) (i : I)
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/-- In the expansions of `F (FreeAlgebra.ι ℂ i * koszulOrder q le (ofList r x))`
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the ter multiplying `F ((koszulOrder q le) (ofList (optionEraseZ r i n) x))`. -/
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def superCommuteCenterOrder (r : List 𝓕) (i : 𝓕)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ[ℂ] A)
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(F : FreeAlgebra ℂ 𝓕 →ₐ[ℂ] A)
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(n : Option (Fin r.length)) : A :=
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match n with
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| none => 1
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@ -222,24 +217,21 @@ def superCommuteCenterOrder {I : Type}
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(FreeAlgebra.ι ℂ (r.get n)))
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@[simp]
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lemma superCommuteCenterOrder_none {I : Type}
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(q : I → FieldStatistic) (r : List I) (i : I)
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lemma superCommuteCenterOrder_none (r : List 𝓕) (i : 𝓕)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ[ℂ] A) :
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(F : FreeAlgebra ℂ 𝓕 →ₐ[ℂ] A) :
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superCommuteCenterOrder q r i F none = 1 := by
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simp [superCommuteCenterOrder]
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open HepLean.List
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lemma le_all_mul_koszulOrder_ofList_expand {I : Type}
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(q : I → FieldStatistic) (r : List I) (x : ℂ) (le1 : I → I→ Prop) [DecidableRel le1]
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[IsTotal I le1] [IsTrans I le1]
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(i : I) (hi : ∀ (j : I), le1 j i)
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lemma le_all_mul_koszulOrder_ofList_expand [IsTotal 𝓕 le] [IsTrans 𝓕 le] (r : List 𝓕) (x : ℂ)
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(i : 𝓕) (hi : ∀ (j : 𝓕), le j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ[ℂ] A) [OperatorMap q le1 F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder q le1 (ofList r x)) =
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(F : FreeAlgebra ℂ 𝓕 →ₐ[ℂ] A) [OperatorMap q le F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder q le (ofList r x)) =
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∑ n, superCommuteCenterOrder q r i F n *
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F ((koszulOrder q le1) (ofList (optionEraseZ r i n) x)) := by
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F ((koszulOrder q le) (ofList (optionEraseZ r i n) x)) := by
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rw [le_all_mul_koszulOrder_ofList]
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conv_lhs =>
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rhs
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@ -255,18 +247,18 @@ lemma le_all_mul_koszulOrder_ofList_expand {I : Type}
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rfl
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exact fun j => hi j
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lemma le_all_mul_koszulOrder_ofListLift_expand {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → FieldStatistic) (r : List I) (x : ℂ)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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[IsTotal (Σ i, f i) le1] [IsTrans (Σ i, f i) le1]
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(i : (Σ i, f i)) (hi : ∀ (j : (Σ i, f i)), le1 j i)
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lemma le_all_mul_koszulOrder_ofListLift_expand {f : 𝓕 → Type} [∀ i, Fintype (f i)]
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(r : List 𝓕) (x : ℂ)
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(le : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le]
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[IsTotal (Σ i, f i) le] [IsTrans (Σ i, f i) le]
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(i : (Σ i, f i)) (hi : ∀ (j : (Σ i, f i)), le j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F] :
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F (ofList [i] 1 * koszulOrder (fun i => q i.1) le1 (ofListLift f r x)) =
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F ((koszulOrder (fun i => q i.fst) le1) (ofList [i] 1 * ofListLift f r x)) +
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le F] :
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F (ofList [i] 1 * koszulOrder (fun i => q i.1) le (ofListLift f r x)) =
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F ((koszulOrder (fun i => q i.fst) le) (ofList [i] 1 * ofListLift f r x)) +
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∑ n : (Fin r.length), superCommuteCoef q [r.get n] (List.take (↑n) r) •
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F (((superCommute fun i => q i.fst) (ofList [i] 1)) (ofListLift f [r.get n] 1)) *
|
||||
F ((koszulOrder (fun i => q i.fst) le1) (ofListLift f (r.eraseIdx ↑n) x)) := by
|
||||
F ((koszulOrder (fun i => q i.fst) le) (ofListLift f (r.eraseIdx ↑n) x)) := by
|
||||
match r with
|
||||
| [] =>
|
||||
simp only [map_mul, List.length_nil, Finset.univ_eq_empty, List.get_eq_getElem, List.take_nil,
|
||||
|
|
@ -332,7 +324,6 @@ lemma le_all_mul_koszulOrder_ofListLift_expand {I : Type} {f : I → Type} [∀
|
|||
rhs
|
||||
intro a0
|
||||
rw [← Finset.mul_sum]
|
||||
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro a0
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue