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

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Real and imaginary kac moody roots

Definition

For the root system of $g (A)$ , define $Δ^{r e} = W Π$ , where $Π = {α_{1}, \dots, α_{n}}$ . Define $Δ^{i m} = Δ ∖ Δ^{r e}$ . These are the real and imaginary roots. Each inherits its positive or negative sign from $Δ \subset Q^{+} \cup (- Q^{+})$ .

By The weyl group preserves roots and root multiplicities, $W Π$ consists of roots. The two classes partition $Δ$ and exclude zero. The labels denote orbit membership; they do not define roots by the sign of a squared length, and make sense without a symmetrizer.

Sources

Source comparison: Kleshchev, §5.1, pp.68–69, and §5.3, pp.73–74.

Depends on

The weyl group preserves roots and root multiplicities

Used by

Imaginary root spaces need not have multiplicity one Counterexample
A symmetrizable indefinite rank two gcm Example
Real root spaces are one dimensional sl2 roots Proposition

Dependency tree · two levels

4 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 — §5.1, pp.68–69, and §5.3, pp.73–74 (standard reference, not scraped)