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Killing-dual vector of a root

Definition

Assume the Axiom of Choice. Let $h$ be a Cartan subalgebra of a finite-dimensional complex semisimple Lie algebra $g$ , with root set $Φ$ (Root and root space), and let $B$ be the Killing form (Killing form). Since $B ∣_{h}$ is nondegenerate by Orthogonality of root spaces and nondegeneracy on the Cartan subalgebra, the map $h \to h^{*}$ , $H \mapsto B (H, \cdot)$ , is a linear isomorphism; for $α \in Φ$ the unique vector

$H_{α} \in h, B (H_{α}, H) = α (H) for all H \in h,$

is the Killing-dual vector of the root $α$ . For $α \neq 0$ it is nonzero, since a nonzero functional cannot be represented by the zero vector under an isomorphism.

Depends on

Used by

The only scalar multiples of a root that are roots are plus or minus the root Corollary
Cayley transform of a theta-stable Cartan subalgebra Definition
Coroot of a Lie-algebra root Definition
The Killing form identifies roots with coroot directions Example
The root sl₂ triple inside slₙ Example
Chevalley basis and real structure constants Lemma
The Killing length of a root is nonzero Lemma
The bracket of opposite root spaces is the root line Proposition
The roots form a reduced crystallographic Euclidean root system Proposition
Cayley transforms connect theta-stable Cartans in the classification Theorem
Existence and uniqueness up to isomorphism of the split real form Theorem
Roots of a complex semisimple Lie algebra form a reduced crystallographic root system Theorem
The root sl₂ triple Theorem

Dependency tree · two levels

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)