Theorems · Theorem · nonassociative algebras

LieDerivation.IsKilling.ad_mem_orthogonal_of_mem_orthogonal

∀ {R : Type u_1} {L : Type u_2} [inst : Field R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  {D : LieDerivation R L L},
  D ∈ (killingForm R (LieDerivation R L L)).orthogonal (LieDerivation.ad R L).range.toSubmodule →
    ∀ (x : L),
      (LieDerivation.ad R L) (D x) ∈
        (killingForm R (LieDerivation R L L)).orthogonal (LieDerivation.ad R L).range.toSubmodule

If a derivation D is in the Killing orthogonal of the range of the adjoint action, then, for any x : L, ad (D x) is also in this orthogonal.

Defined in

Mathlib.Algebra.Lie.Derivation.Killing

Cited by

1 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLieRingLieAlgebra