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namespace OAI
/-! Vanishing normal cocycles for skew Lie triples. -/
namespace BFSSNormal
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
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⟩
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
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
end OAI
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