DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Regular element and rank

Definition

Let $g$ be a finite-dimensional complex Lie algebra (Lie algebras over a field), and for $x \in g$ write $g^{x} = \ker ({ad}_{x}) = {y \in g : [x, y] = 0}$ using ${ad}_{x} (y) = [x, y]$ from Derivations of Lie algebras. The numbers $\dim g^{x}$ are natural numbers bounded by $\dim g$ , so the set of values attained has a least element; it is denoted $rank (g)$ .

An element $x \in g$ is regular if $\dim g^{x} = rank (g)$ , and regular semisimple if in addition ${ad}_{x}$ is a semisimple endomorphism in the sense of Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms. Thus every regular semisimple element is regular and has semisimple adjoint operator, and for $g = 0$ the single element $0$ is regular semisimple with $rank (0) = 0$ .

Depends on

Used by

Regular and singular diagonal elements of slₙ Example
Every element of a complex semisimple Lie algebra is semisimple False statement
Dimension formula from roots Proposition
Centralizer of a regular semisimple element is Cartan Theorem

Dependency tree · two levels

12 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)