DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Simple reflections and the kac moody weyl group

Definition

For a minimal realization, put $s_{i} (λ) = λ - λ (h_{i}) α_{i}$ on $h^{*}$ . The Weyl group is $W = ⟨ s_{1}, \dots, s_{n} ⟩ \subset GL (h^{*})$ . Its dual action is $s_{i} (h) = h - α_{i} (h) h_{i}$ .

By Realization of a generalized cartan matrix, $α_{i} (h_{i}) = 2$ . Hence $(s_{i} λ) (h_{i}) = - λ (h_{i})$ and $s_{i}^{2} λ = λ$ . Furthermore $s_{i} α_{j} = α_{j} - a_{i j} α_{i}$ , so $s_{i}$ preserves the lattice in Kac Moody root lattice height and positive cone. The two actions are dual since $λ (s_{i} h) = (s_{i} λ) (h)$ . This definition requires neither a finite group nor a Coxeter presentation.

Sources

Source comparison: Kleshchev, §3.2, pp.39–42.

Depends on

Used by

Dependency tree · two levels

3 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

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §3.2, pp.39–42 (standard reference, not scraped)