DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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Nilpotent element in a finite-dimensional Lie algebra

Definition

Let $g$ be a finite-dimensional complex Lie algebra and $f \in g$ . The element $f$ is nilpotent if its adjoint endomorphism $ad f : g \to g$ , $x \mapsto [f, x]$ , is nilpotent; explicitly, $(ad f)^{N} = 0$ for some integer $N \geq 1$ . This definition concerns the adjoint representation and does not assert nilpotence in every representation.

Facts & Assumptions

Given: $g$ and its Lie bracket as in Finite-dimensional simple complex Lie algebra, and an element $f \in g$ .

[F1]

Finite-dimensional simple complex Lie algebra: $g$ is a finite-dimensional complex Lie algebra, so $x \mapsto [f, x]$ is a linear endomorphism of $g$ .

Proof

1.1givenF1algebra

Bilinearity of the Lie bracket makes $x \mapsto [f, x]$ linear, and closure of the bracket in $g$ makes its codomain $g$ . Thus the iterates $(ad f)^{N}$ are defined endomorphisms.

2.1givenF1step 1.1algebra∎

By definition, the displayed vanishing condition is exactly nilpotence of this endomorphism; it imposes no condition on a representation other than the adjoint representation.

Nilpotent element in a finite-dimensional Lie algebra

Definition

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources