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

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.

Kac moody algebra associated to a gcm

Definition

The Kac–Moody algebra of $A$ is $g (A) = \tilde{g} (A) / r$ , using the largest Cartan-disjoint ideal constructed in The sum of triangularly disjoint graded ideals is disjoint from h. We retain the names $h, e_{i}, f_{i}$ for their images and put $n^{\pm} = {\tilde{n}}^{\pm} / r^{\pm}$ .

The Cartan embeds because $r \cap h = 0$ . The sign-changing involution preserves $r$ by its defining largest-ideal property, and descends. Every nonzero ideal of $g (A)$ meets $h$ nontrivially: otherwise its inverse image would be a larger Cartan-disjoint ideal in $\tilde{g}$ . This does not assert simplicity for singular or decomposable matrices.

Sources

Source comparison: Kleshchev, Definition 1.4.1, p.16.

Depends on

The sum of triangularly disjoint graded ideals is disjoint from h

Used by

Imaginary root spaces need not have multiplicity one Counterexample
Nonsingular indecomposable Kac–Moody algebras are simple Lemma
Serre elements vanish before Serre generation Lemma
The opposite simple centralizer in a Kac Moody half vanishes Lemma
Kac moody root spaces are finite dimensional Proposition

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 — Definition 1.4.1, p.16 (standard reference, not scraped)