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.

Generalized cartan matrix

Definition

A generalized Cartan matrix (GCM) is an integer matrix $A = (a_{i j})_{i, j \in I}$ , where $I = {1, \dots, n}$ and $n \geq 1$ , such that $a_{i i} = 2$ , $a_{i j} \leq 0$ for $i \neq j$ , and $a_{i j} = 0 ⟺ a_{j i} = 0$ . It is indecomposable if there is no partition $I = J ⊔ K$ into nonempty sets with all entries between $J$ and $K$ zero.

Sources

Source comparison: Kleshchev, §1.2, p.10.

Used by

A matrix with one zero off diagonal is not a gcm Counterexample
Realization of a generalized cartan matrix Definition
Symmetrizable generalized cartan matrix Definition
Nonsingular indecomposable Kac–Moody algebras are simple Lemma
Finite affine indefinite trichotomy for indecomposable gcms Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §1.2, p.10 (standard reference, not scraped)