Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra

∀ (R : Type u_1) (L : Type u_3) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [LieAlgebra.IsKilling R L] [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra],
  LinearMap.ker ((killingForm R L).restrict H.toSubmodule) = ⊥

If the Killing form of a Lie algebra is non-singular, it remains non-singular when restricted to a Cartan subalgebra.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

1 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraLieAlgebra.IsKillingIsNoetherianIsArtinianLieSubalgebra.IsCartanSubalgebra