DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-09-23 (gpt-6-sol)

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The Killing-dual vector attached to a root

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $g$ be a finite-dimensional complex semisimple Lie algebra, let $h$ be a Cartan subalgebra in the nilpotent self-normalizing sense, and let $α \in h^{*}$ be a root. By Cartan subalgebras are exactly maximal toral subalgebras, $h$ is maximal toral. Thus Finite semisimple Cartan, root and string structure applies and the Killing form restricts nondegenerately to $h$ . Therefore there is a unique vector $H_{α} \in h$ such that

$α (h) = B (H_{α}, h) (h \in h) .$

This vector is the Killing-dual vector attached to the root $α$ .

Depends on

Used by

Root reflections and the Weyl group action Definition
The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
Opposite root spaces bracket to the Killing-dual line Proposition
Generic irreducibility and the exceptional parameter lattice Theorem
The root set is a reduced crystallographic root system Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)