Theorems · Theorem · nonassociative algebras

LieAlgebra.InvariantForm.isSemisimple_of_nondegenerate

∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : LieRing L] [inst_2 : LieAlgebra K L] [Module.Finite K L]
  (Φ : LinearMap.BilinForm K L),
  Φ.Nondegenerate →
    LinearMap.BilinForm.lieInvariant L Φ →
      Φ.IsRefl → (∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I) → LieAlgebra.IsSemisimple K L

A finite-dimensional Lie algebra over a field is semisimple if it does not have non-trivial abelian ideals and it admits a non-degenerate reflexive invariant bilinear form. Here a form is invariant if it is compatible with the Lie bracket: Φ ⁅x, y⁆ z = Φ x ⁅y, z⁆.

Defined in

Mathlib.Algebra.Lie.InvariantForm

Cited by

0 results in Mathlib

Foundations

Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLieRingLieAlgebraModule.Finite