| import Mathlib |
| import OAI.Algebra.MarkedTensor.AdjointCoordinates |
|
|
| namespace OAI |
|
|
| /-! Nine polynomial conjugators spanning every adjoint-coordinate endomorphism. -/ |
|
|
| noncomputable section |
| open scoped BigOperators commutatorElement |
| open Matrix |
|
|
| namespace MarkedTensor |
| namespace Adjoint |
|
|
| variable {K : Type*} [Field K] |
|
|
| abbrev Mat₂ (K : Type*) := Matrix (Fin 2) (Fin 2) K |
| abbrev Mat₃ (K : Type*) := Matrix (Fin 3) (Fin 3) K |
| abbrev Vec₃ (K : Type*) := Fin 3 → K |
|
|
| abbrev mat (g : SpecialLinearGroup (Fin 2) K) : Mat₂ K := g |
|
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| |
| def traceless (v : Vec₃ K) : Mat₂ K := !![v 1, v 0; v 2, -v 1] |
|
|
| def extract (A : Mat₂ K) : Vec₃ K := ![A 0 1, A 0 0, A 1 0] |
|
|
| def tracelessLM : Vec₃ K →ₗ[K] Mat₂ K where |
| toFun := traceless |
| map_add' x y := by ext i j; fin_cases i <;> fin_cases j <;> simp [traceless, add_comm] |
| map_smul' c x := by ext i j; fin_cases i <;> fin_cases j <;> simp [traceless] |
|
|
| def extractLM : Mat₂ K →ₗ[K] Vec₃ K where |
| toFun := extract |
| map_add' x y := by ext i; fin_cases i <;> simp [extract] |
| map_smul' c x := by ext i; fin_cases i <;> simp [extract] |
|
|
| def conjugationLM (g : SpecialLinearGroup (Fin 2) K) : Mat₂ K →ₗ[K] Mat₂ K where |
| toFun A := mat g * A * mat g⁻¹ |
| map_add' A B := by simp only [mul_add, add_mul] |
| map_smul' c A := by simp only [mul_smul_comm, smul_mul_assoc, RingHom.id_apply] |
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| |
| def adjoint (g : SpecialLinearGroup (Fin 2) K) : Vec₃ K →ₗ[K] Vec₃ K := |
| extractLM ∘ₗ conjugationLM g ∘ₗ tracelessLM |
|
|
| lemma traceless_extract (v : Vec₃ K) : extract (traceless v) = v := by |
| ext i; fin_cases i <;> simp [extract, traceless] |
|
|
| lemma extract_traceless (A : Mat₂ K) (hA : Matrix.trace A = 0) : |
| traceless (extract A) = A := by |
| have hdiag : A 0 0 + A 1 1 = 0 := by simpa [Matrix.trace, Fin.sum_univ_two] using hA |
| ext i j; fin_cases i <;> fin_cases j <;> simp [traceless, extract] |
| exact neg_eq_of_add_eq_zero_right hdiag |
|
|
| lemma traceless_adjoint (g : SpecialLinearGroup (Fin 2) K) (v : Vec₃ K) : |
| traceless (adjoint g v) = mat g * traceless v * mat g⁻¹ := by |
| apply extract_traceless |
| change Matrix.trace (mat g * traceless v * mat g⁻¹) = 0 |
| rw [Matrix.trace_mul_cycle] |
| have hi : mat g⁻¹ * mat g = 1 := |
| congrArg mat (inv_mul_cancel g) |
| rw [hi, one_mul] |
| simp [Matrix.trace, Fin.sum_univ_two, traceless] |
|
|
| |
| def conjugator (x y : K) : SpecialLinearGroup (Fin 2) K := |
| ⟨!![1 + x*y, x; y, 1], by simp [Matrix.det_fin_two]⟩ |
|
|
| lemma conjugator_inv (x y : K) : |
| mat (conjugator x y)⁻¹ = |
| !![1, -x; -y, 1 + x*y] := by |
| change Matrix.adjugate (!![1 + x*y, x; y, 1] : Mat₂ K) = _ |
| rw [Matrix.adjugate_fin_two] |
| rfl |
|
|
| |
| def adjointMatrix (x y : K) : Mat₃ K := |
| !![(1 + x*y)^2, -2*x*(1 + x*y), -x^2; |
| -y*(1 + x*y), 1 + 2*x*y, x; |
| -y^2, 2*y, 1] |
|
|
| lemma toMatrix_adjoint (x y : K) : |
| LinearMap.toMatrix' (adjoint (conjugator x y)) = adjointMatrix x y := by |
| ext i j |
| change extract (mat (conjugator x y) * traceless (Pi.single j 1) * |
| mat (conjugator x y)⁻¹) i = adjointMatrix x y i j |
| rw [conjugator_inv] |
| fin_cases i <;> fin_cases j <;> |
| simp [extract, traceless, mat, conjugator, adjointMatrix, |
| Matrix.mul_apply, Fin.sum_univ_two] <;> ring |
|
|
| |
| def coefficient : Fin 3 → Fin 3 → Mat₃ K := ![ |
| ![1, !![0,0,0; -1,0,0; 0,2,0], !![0,0,0; 0,0,0; -1,0,0]], |
| ![!![0,-2,0; 0,0,1; 0,0,0], !![2,0,0; 0,2,0; 0,0,0], |
| !![0,0,0; -1,0,0; 0,0,0]], |
| ![!![0,0,-1; 0,0,0; 0,0,0], !![0,-2,0; 0,0,0; 0,0,0], |
| !![1,0,0; 0,0,0; 0,0,0]]] |
|
|
| @[simp] lemma coefficient00 : coefficient (K := K) 0 0 = 1 := rfl |
| @[simp] lemma coefficient01 : coefficient (K := K) 0 1 = |
| !![0,0,0; -1,0,0; 0,2,0] := rfl |
| @[simp] lemma coefficient02 : coefficient (K := K) 0 2 = |
| !![0,0,0; 0,0,0; -1,0,0] := rfl |
| @[simp] lemma coefficient10 : coefficient (K := K) 1 0 = |
| !![0,-2,0; 0,0,1; 0,0,0] := rfl |
| @[simp] lemma coefficient11 : coefficient (K := K) 1 1 = |
| !![2,0,0; 0,2,0; 0,0,0] := rfl |
| @[simp] lemma coefficient12 : coefficient (K := K) 1 2 = |
| !![0,0,0; -1,0,0; 0,0,0] := rfl |
| @[simp] lemma coefficient20 : coefficient (K := K) 2 0 = |
| !![0,0,-1; 0,0,0; 0,0,0] := rfl |
| @[simp] lemma coefficient21 : coefficient (K := K) 2 1 = |
| !![0,-2,0; 0,0,0; 0,0,0] := rfl |
| @[simp] lemma coefficient22 : coefficient (K := K) 2 2 = |
| !![1,0,0; 0,0,0; 0,0,0] := rfl |
|
|
| def quadratic {V : Type*} [AddCommGroup V] [Module K V] (a b c : V) (t : K) : V := |
| a + t • b + t^2 • c |
|
|
| lemma quadratic_coefficients_mem [CharZero K] {V : Type*} [AddCommGroup V] [Module K V] |
| (W : Submodule K V) (a b c : V) {s : K} (hs : s ≠ 0) |
| (h0 : quadratic a b c (0 : K) ∈ W) (h1 : quadratic a b c s ∈ W) |
| (h2 : quadratic a b c (2*s) ∈ W) : a ∈ W ∧ b ∈ W ∧ c ∈ W := by |
| have ha : a ∈ W := by simpa [quadratic] using h0 |
| have he : quadratic a b c (2*s) - (2 : K) • quadratic a b c s + a = |
| (2*s^2) • c := by unfold quadratic; module |
| have hc' : (2*s^2) • c ∈ W := he ▸ W.add_mem (W.sub_mem h2 (W.smul_mem 2 h1)) ha |
| have hc : c ∈ W := by |
| have h := W.smul_mem (2*s^2)⁻¹ hc' |
| simpa only [smul_smul, inv_mul_cancel₀ (mul_ne_zero (by norm_num : (2 : K) ≠ 0) |
| (pow_ne_zero 2 hs)), one_smul] using h |
| have hb' : s • b ∈ W := by |
| convert W.sub_mem (W.sub_mem h1 ha) (W.smul_mem (s^2) hc) using 1 |
| unfold quadratic |
| module |
| have hb : b ∈ W := by |
| have h := W.smul_mem s⁻¹ hb' |
| simpa only [smul_smul, inv_mul_cancel₀ hs, one_smul] using h |
| exact ⟨ha, hb, hc⟩ |
|
|
| def rowPolynomial (p : Fin 3) (y : K) : Mat₃ K := |
| quadratic (coefficient p 0) (coefficient p 1) (coefficient p 2) y |
|
|
| lemma expansion (x y : K) : adjointMatrix x y = |
| quadratic (rowPolynomial 0 y) (rowPolynomial 1 y) (rowPolynomial 2 y) x := by |
| ext i j; fin_cases i <;> fin_cases j <;> |
| simp [adjointMatrix, quadratic, rowPolynomial, coefficient] <;> ring |
|
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| variable [CharZero K] |
|
|
| lemma coefficients_mem {W : Submodule K (Mat₃ K)} {s : K} (hs : s ≠ 0) |
| (h : ∀ a b : Fin 3, adjointMatrix ((a : K)*s) ((b : K)*s) ∈ W) |
| (p q : Fin 3) : coefficient (K := K) p q ∈ W := by |
| have hr (b : Fin 3) : ∀ p : Fin 3, rowPolynomial p ((b : K)*s) ∈ W := by |
| have hh := quadratic_coefficients_mem W |
| (rowPolynomial 0 ((b : K)*s)) (rowPolynomial 1 ((b : K)*s)) |
| (rowPolynomial 2 ((b : K)*s)) hs |
| (by simpa only [← expansion, Fin.val_zero, Nat.cast_zero, zero_mul] using h 0 b) |
| (by simpa only [← expansion, Fin.val_one, Nat.cast_one, one_mul] using h 1 b) |
| (by simpa only [← expansion, Fin.val_two, Nat.cast_ofNat] using h 2 b) |
| intro p; fin_cases p |
| · exact hh.1 |
| · exact hh.2.1 |
| · exact hh.2.2 |
| have hh := quadratic_coefficients_mem W (coefficient p 0) (coefficient p 1) |
| (coefficient p 2) hs |
| (by simpa only [rowPolynomial, Fin.val_zero, Nat.cast_zero, zero_mul] using hr 0 p) |
| (by simpa only [rowPolynomial, Fin.val_one, Nat.cast_one, one_mul] using hr 1 p) |
| (by simpa only [rowPolynomial, Fin.val_two, Nat.cast_ofNat] using hr 2 p) |
| fin_cases q |
| · exact hh.1 |
| · exact hh.2.1 |
| · exact hh.2.2 |
|
|
| lemma span_eq_top_of_coefficients {W : Submodule K (Mat₃ K)} |
| (h : ∀ p q : Fin 3, coefficient p q ∈ W) : W = ⊤ := by |
| have hu (i j : Fin 3) : Matrix.single i j (1 : K) ∈ W := by |
| fin_cases i <;> fin_cases j |
| · convert h 2 2 using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.smul_mem (-1/2 : K) (h 2 1) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.neg_mem (h 2 0) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.neg_mem (h 1 2) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.sub_mem (W.smul_mem (1/2 : K) (h 1 1)) (h 2 2) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.sub_mem (h 1 0) (h 2 1) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.neg_mem (h 0 2) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.smul_mem (1/2 : K) (W.sub_mem (h 0 1) (h 1 2)) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| · convert W.sub_mem (h 0 0) (W.smul_mem (1/2 : K) (h 1 1)) using 1 |
| ext row column |
| fin_cases row <;> fin_cases column <;> norm_num [Matrix.single] |
| apply top_unique |
| intro B _ |
| rw [Matrix.matrix_eq_sum_single B] |
| apply W.sum_mem; intro i _ |
| apply W.sum_mem; intro j _ |
| simpa only [Matrix.smul_single, smul_eq_mul, mul_one] using W.smul_mem (B i j) (hu i j) |
|
|
| |
| lemma nine_matrix_span {s : K} (hs : s ≠ 0) : |
| Submodule.span K (Set.range (fun ab : Fin 3 × Fin 3 => |
| adjointMatrix ((ab.1 : K)*s) ((ab.2 : K)*s))) = ⊤ := by |
| apply span_eq_top_of_coefficients |
| apply coefficients_mem hs |
| intro a b |
| exact Submodule.subset_span ⟨(a,b), rfl⟩ |
|
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| |
|
|
| theorem vertex_adjoint_span {s : K} (hs : s ≠ 0) : |
| Submodule.span K (Set.range (fun ab : Fin 3 × Fin 3 => |
| adjoint (conjugator ((ab.1 : K)*s) ((ab.2 : K)*s)))) = ⊤ := by |
| have ht : Submodule.span K (Set.range (fun ab : Fin 3 × Fin 3 => |
| LinearMap.toMatrix' (adjoint (conjugator ((ab.1 : K)*s) ((ab.2 : K)*s))))) = ⊤ := by |
| simpa only [toMatrix_adjoint] using nine_matrix_span hs |
| have hh := congrArg (Submodule.map Matrix.toLin'.toLinearMap) ht |
| simpa only [Submodule.map_span, ← Set.range_comp, Submodule.map_top, |
| LinearEquiv.range, Function.comp_def, LinearEquiv.coe_coe, |
| Matrix.toLin'_toMatrix'] using hh |
|
|
| end Adjoint |
| end MarkedTensor |
| end |
|
|
| end OAI |
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|
