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 Cartan data for quantum groups Definition
Symmetrizable generalized cartan matrix Definition
The cyclic affine type A Cartan datum and its residues Definition
Nonsingular indecomposable Kac–Moody algebras are simple Lemma
The affine simple root alpha zero is delta minus the highest root Lemma
Finite affine indefinite trichotomy for indecomposable gcms Theorem
The formal quantum Serre half embeds in the shuffle algebra and is degreewise free Theorem
Uniqueness of the i-string decomposition of an integrable module 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)