instInvolutiveNegSubtypeMemLieSubalgebra

English

There exists an abelian structure on a Cartan subalgebra under Killing assumptions.

Русский

Существуют абелевыess свойства Cartan-подалгебры при предположениях Killing.

LaTeX

$$$ IsLieAbelian(H) $$$

Lean4

instance : InvolutiveNeg (Weight K H L)
    where
  neg
    α :=
    ⟨-α, by
      by_cases hα : α.IsZero
      · convert α.genWeightSpace_ne_bot; rw [hα, neg_zero]
      · intro e
        obtain ⟨x, hx, x_ne0⟩ := α.exists_ne_zero
        have :=
          mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_neg K L H hx
            (fun y hy ↦ by rw [rootSpace, e] at hy; rw [hy, map_zero])
        rw [ker_killingForm_eq_bot] at this
        exact x_ne0 this⟩
  neg_neg α := by ext; simp

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