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

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Kac Moody root lattice height and positive cone

Definition

For a realization as in Realization of a generalized cartan matrix, set $Q = ⨁_{i = 1}^{n} Z α_{i}$ and $Q^{+} = ⨁_{i = 1}^{n} Z_{\geq 0} α_{i}$ . Define $ht (\sum_{i} k_{i} α_{i}) = \sum_{i} k_{i}$ . For $λ, μ \in h^{*}$ write $μ \leq λ$ if $λ - μ \in Q^{+}$ .

Independence of the simple roots makes coordinates unique. Thus $Q^{+} \cap (- Q^{+}) = {0}$ , which proves antisymmetry of the order; closure under addition proves transitivity, and $0 \in Q^{+}$ proves reflexivity. Positive roots will belong to $Q^{+} ∖ {0}$ , not all of $Q^{+}$ .

Sources

Source comparison: Kleshchev, end of §1.2, pp.12–13.

Depends on

Realization of a generalized cartan matrix

Used by

Contragredient lie algebra before the maximal ideal quotient Definition
Kac moody category o Definition
Simple reflections and the kac moody weyl group Definition

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

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — end of §1.2, pp.12–13 (standard reference, not scraped)