DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Coroot and dual root system

Definition

Let $Φ \subseteq E$ be a reduced crystallographic root system in the real inner product space $E$ (Reduced crystallographic Euclidean root system). For $α \in Φ$ put $α^{\lor} = \frac{2 α}{(α, α)} \in E .$ The vector $α^{\lor}$ is the coroot of $α$ , and the set $Φ^{\lor} = {α^{\lor} : α \in Φ}$ is the dual root system.

The definitions are well posed because $(α, α) > 0$ for $α \neq 0$ . Two elementary identities, obtained by substituting the definition and using bilinearity of the inner product, are $(α^{\lor}, β) = \frac{2 (β, α)}{(α, α)}, 2 \frac{α^{\lor}}{(α^{\lor}, α^{\lor})} = α,$ for all $α, β \in Φ$ ; in particular $α^{\lor}^{\lor} = α$ , and the Cartan integer $2 (β, α) / (α, α)$ equals the inner product $(α^{\lor}, β)$ . Also $α^{\lor}$ is a positive real multiple of $α$ , so $R α^{\lor} = R α$ and $R α^{\lor} \cap Φ^{\lor} = {\pm α^{\lor}}$ .

Depends on

Reduced crystallographic Euclidean root system

Used by

Cartan matrix of a based root system Definition
Root, coroot, weight, and coweight lattices Definition
Weyl group Definition
Simple roots and fundamental weights of Aₙ Example
Chevalley basis and real structure constants Lemma
Duality exchanges B and C Proposition
The Weyl group is finite and faithful Proposition
The Weyl vector in fundamental coordinates Proposition
Rank-two root-system classification Theorem

Dependency tree · two levels

2 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

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