| import Mathlib |
|
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| namespace OAI |
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| /-! Vanishing normal cocycles for skew Lie triples. -/ |
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| namespace BFSSNormal |
| noncomputable section |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
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| lemma skew_sum_sq_kernel (X Y Z : E →ₗ[ℝ] E) |
| (hX : ∀ u v, inner ℝ (X u) v = -inner ℝ u (X v)) |
| (hY : ∀ u v, inner ℝ (Y u) v = -inner ℝ u (Y v)) |
| (hZ : ∀ u v, inner ℝ (Z u) v = -inner ℝ u (Z v)) |
| (a : E) (ha : X (X a) + Y (Y a) + Z (Z a) = 0) : |
| X a = 0 ∧ Y a = 0 ∧ Z a = 0 := by |
| have hh := congrArg (fun v => inner ℝ a v) ha |
| simp only [inner_add_right, inner_zero_right] at hh |
| have hx := hX a (X a) |
| have hy := hY a (Y a) |
| have hz := hZ a (Z a) |
| rw [real_inner_self_eq_norm_sq] at hx hy hz |
| have h0 : ‖X a‖^2 + ‖Y a‖^2 + ‖Z a‖^2 = 0 := by linarith |
| have hx0 : ‖X a‖ = 0 := by nlinarith [sq_nonneg ‖Y a‖, sq_nonneg ‖Z a‖] |
| have hy0 : ‖Y a‖ = 0 := by nlinarith [sq_nonneg ‖X a‖, sq_nonneg ‖Z a‖] |
| have hz0 : ‖Z a‖ = 0 := by nlinarith [sq_nonneg ‖X a‖, sq_nonneg ‖Y a‖] |
| exact ⟨norm_eq_zero.mp hx0,norm_eq_zero.mp hy0,norm_eq_zero.mp hz0⟩ |
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| lemma cocycle_laplacian (X Y Z : E →ₗ[ℝ] E) |
| (hYX : ∀ u, Y (X u) = X (Y u) - (2:ℝ) • Z u) |
| (hZX : ∀ u, Z (X u) = X (Z u) + (2:ℝ) • Y u) |
| (a b c : E) |
| (hab : X b - Y a = (2:ℝ) • c) |
| (_hbc : Y c - Z b = (2:ℝ) • a) |
| (hca : Z a - X c = (2:ℝ) • b) : |
| X (X a) + Y (Y a) + Z (Z a) = X (X a + Y b + Z c) := by |
| have hya : Y a = X b - (2:ℝ) • c := by rw [←hab]; abel |
| have hza : Z a = X c + (2:ℝ) • b := by rw [←hca]; abel |
| rw [hya,hza,map_sub,map_add,map_smul,map_smul,hYX,hZX] |
| simp only [map_add] |
| module |
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| theorem normal_cocycle_zero (X Y Z : E →ₗ[ℝ] E) |
| (hX : ∀ u v, inner ℝ (X u) v = -inner ℝ u (X v)) |
| (hY : ∀ u v, inner ℝ (Y u) v = -inner ℝ u (Y v)) |
| (hZ : ∀ u v, inner ℝ (Z u) v = -inner ℝ u (Z v)) |
| (hXY : ∀ u, X (Y u) - Y (X u) = (2:ℝ) • Z u) |
| (hYZ : ∀ u, Y (Z u) - Z (Y u) = (2:ℝ) • X u) |
| (hZX : ∀ u, Z (X u) - X (Z u) = (2:ℝ) • Y u) |
| (a b c : E) |
| (hab : X b - Y a = (2:ℝ) • c) |
| (hbc : Y c - Z b = (2:ℝ) • a) |
| (hca : Z a - X c = (2:ℝ) • b) |
| (hperp : X a + Y b + Z c = 0) : a = 0 ∧ b = 0 ∧ c = 0 := by |
| have hYX' (u : E) : Y (X u) = X (Y u) - (2:ℝ) • Z u := by rw [←hXY u]; abel |
| have hZY' (u : E) : Z (Y u) = Y (Z u) - (2:ℝ) • X u := by rw [←hYZ u]; abel |
| have hXZ' (u : E) : X (Z u) = Z (X u) - (2:ℝ) • Y u := by rw [←hZX u]; abel |
| have hZX' (u : E) : Z (X u) = X (Z u) + (2:ℝ) • Y u := by rw [←hZX u]; abel |
| have hXY' (u : E) : X (Y u) = Y (X u) + (2:ℝ) • Z u := by rw [←hXY u]; abel |
| have hYZ' (u : E) : Y (Z u) = Z (Y u) + (2:ℝ) • X u := by rw [←hYZ u]; abel |
| have ha := cocycle_laplacian X Y Z hYX' hZX' a b c hab hbc hca |
| rw [hperp,map_zero] at ha |
| have hp2 : Y b + Z c + X a = 0 := by rw [←hperp]; abel |
| have hp3 : Z c + X a + Y b = 0 := by rw [←hperp]; abel |
| have hb := cocycle_laplacian Y Z X hZY' hXY' b c a hbc hca hab |
| rw [hp2,map_zero] at hb |
| have hc := cocycle_laplacian Z X Y hXZ' hYZ' c a b hca hab hbc |
| rw [hp3,map_zero] at hc |
| obtain ⟨hxa,hya,hza⟩ := skew_sum_sq_kernel X Y Z hX hY hZ a ha |
| obtain ⟨hyb,hzb,hxb⟩ := skew_sum_sq_kernel Y Z X hY hZ hX b hb |
| obtain ⟨hzc,hxc,hyc⟩ := skew_sum_sq_kernel Z X Y hZ hX hY c hc |
| rw [hxb,hya,sub_self] at hab |
| rw [hyc,hzb,sub_self] at hbc |
| rw [hza,hxc,sub_self] at hca |
| exact ⟨(smul_eq_zero.mp hbc.symm).resolve_left (by norm_num), |
| (smul_eq_zero.mp hca.symm).resolve_left (by norm_num), |
| (smul_eq_zero.mp hab.symm).resolve_left (by norm_num)⟩ |
| end |
| end BFSSNormal |
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| end OAI |
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