Theorems · Theorem · nonassociative algebras

LieModule.isLieAbelian_of_ker_traceForm_eq_bot

∀ (R : Type u_1) (L : Type u_3) (M : Type u_4) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [inst_6 : LieModule R L M]
  [LieRing.IsNilpotent L] [IsDomain R] [Module.Free R M] [Module.Finite R M],
  LinearMap.ker (LieModule.traceForm R L M) = ⊥ → IsLieAbelian L

A nilpotent Lie algebra with a representation whose trace form is non-singular is Abelian.

Defined in

Mathlib.Algebra.Lie.TraceForm

Cited by

0 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRing.IsNilpotentIsDomainModule.FreeModule.Finite