DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Regular root hyperplanes

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $h$ be a Cartan subalgebra of a finite-dimensional complex semisimple Lie algebra $g$ with finite root set $Φ$ (Root and root space, Roots of a complex semisimple Lie algebra form a reduced crystallographic root system). For each root $α$ the root hyperplane is $\ker α = {H \in h : α (H) = 0}$ , a proper subspace of $h$ because $α \neq 0$ . The regular set of $h$ is the complement

$h_{r e g} = {H \in h : α (H) \neq 0 for every α \in Φ} = h ∖ ⋃_{α \in Φ} \ker α .$

It is the complement in $h$ of a finite union of hyperplanes. Elements of $h_{r e g}$ are called regular elements of $h$ .

Depends on

Used by

Regular elements form a dense Zariski-open subset of a Cartan subalgebra Corollary
Regular and singular diagonal elements of slₙ Example
Centralizer dimension from vanishing roots Proposition

Dependency tree · two levels

17 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)