range_ad_eq_top — Mathlib

English

The range of the adjoint action is the whole Lie algebra of derivations considered, i.e., it equals the top subalgebra.

Русский

Диапазон adjoint совпадает с всей структурой, т.е. равен топ-подпростобу.

LaTeX

$$Range( ad_{R,L} ) = 𝕀 = ⊤.$$

Lean4

/-- The range of the adjoint action on a finite-dimensional Killing Lie algebra is full. -/
@[simp]
theorem range_ad_eq_top : 𝕀 = ⊤ := by
  rw [← LieSubalgebra.toSubmodule_inj]
  apply
    LinearMap.BilinForm.eq_top_of_restrict_nondegenerate_of_orthogonal_eq_bot (LieModule.traceForm_isSymm R 𝔻 𝔻).isRefl
      (killingForm_restrict_range_ad_nondegenerate R L)
  refine (Submodule.eq_bot_iff _).mpr fun D hD ↦ ext fun x ↦ ?_
  simpa using ad_mem_ker_killingForm_ad_range_of_mem_orthogonal hD x

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