Traces and nilpotent endomorphisms in characteristic zero

Over a field of characteristic zero, an endomorphism f of a finite-dimensional vector space is nilpotent exactly when the traces of all its positive powers vanish. One direction holds over any reduced ring. For the other, factor the characteristic polynomial as X ^ m * q with q(0) ≠ 0 and choose a Bézout identity a * X ^ (m + 1) + b * q = 1. Then a(f) * f ^ (m + 1) is an idempotent polynomial in f with zero constant term. Its trace is its rank and also vanishes, so it is zero, q(f) is invertible, and f ^ m = 0.

Two consequences for a pair of endomorphisms A, B with ⁅A, ⁅A, B⁆⁆ = 0 follow.

• Jacobson's lemma: ⁅A, B⁆ is nilpotent. Indeed ⁅A, B⁆ ^ (k + 1) is the commutator of A with ⁅A, B⁆ ^ k * B, because ⁅A, B⁆ commutes with A, so all these traces vanish.
• If moreover A is nilpotent, then trace (A * B) = 0. The subspaces V i = ker (A ^ i) exhaust the space, A maps V (i + 1) into V i, and B maps V i into V (i + 1). Commuting the two restricted maps under the trace, and using that ⁅A, B⁆ is nilpotent on each V i, shows that the trace of A * B on V i does not depend on i; it vanishes on V 0 = 0.

Applied to the adjoint representation, the last statement says that an ad-nilpotent element is Killing-orthogonal to every element that its adjoint action kills twice. This is the input to the construction of the neutral element of an sl₂-triple in the Jacobson–Morozov theorem.

Main results

• LinearMap.isNilpotent_iff_forall_trace_pow_eq_zero: in characteristic zero, an endomorphism is nilpotent if and only if all traces of its positive powers vanish.
• LinearMap.isNilpotent_lie_of_lie_lie_eq_zero: Jacobson's lemma.
• LinearMap.trace_mul_eq_zero_of_isNilpotent_of_lie_lie_eq_zero: if A is nilpotent and ⁅A, ⁅A, B⁆⁆ = 0, then trace (A * B) = 0.

References

• N. Jacobson, Rational methods in the theory of Lie algebras, Ann. of Math. 36 (1935), 875–881, for the nilpotence of ⁅A, B⁆.
• N. Jacobson, Lie Algebras, Interscience (1962), Chapter III, §11, for its use in the proof of the Jacobson–Morozov theorem.

theorem LinearMap.isNilpotent_iff_forall_trace_pow_eq_zero {K : Type u_1} {M : Type u_2} [Field K] [AddCommGroup M] [Module K M] [FiniteDimensional K M] [CharZero K] {f : Module.End K M} : IsNilpotent f ↔ ∀ (k : ℕ), 0 < k → (trace K M) (f ^ k) = 0

Trace criterion for nilpotence. Over a field of characteristic zero, an endomorphism of a finite-dimensional vector space is nilpotent if and only if the traces of all its positive powers vanish.

theorem LinearMap.isNilpotent_lie_of_lie_lie_eq_zero {K : Type u_1} {M : Type u_2} [Field K] [AddCommGroup M] [Module K M] [FiniteDimensional K M] [CharZero K] {A B : Module.End K M} (h : ⁅A, ⁅A, B⁆⁆ = 0) : IsNilpotent ⁅A, B⁆

Jacobson's lemma on commutators. In characteristic zero, if A commutes with ⁅A, B⁆, then ⁅A, B⁆ is nilpotent: each of its positive powers is a commutator with A, hence traceless.

theorem LinearMap.trace_mul_eq_zero_of_isNilpotent_of_lie_lie_eq_zero {K : Type u_1} {M : Type u_2} [Field K] [AddCommGroup M] [Module K M] [FiniteDimensional K M] [CharZero K] {A B : Module.End K M} (hA : IsNilpotent A) (h : ⁅A, ⁅A, B⁆⁆ = 0) : (trace K M) (A * B) = 0

The trace of A * B vanishes when A is nilpotent and commutes with ⁅A, B⁆. In characteristic zero, the trace of A * B on ker (A ^ i) is independent of i, since B raises and A lowers the index by one and ⁅A, B⁆ is nilpotent by Jacobson's lemma.