DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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.

Affine node-zero fundamental weight

Definition

Let $\hat{h} = h \oplus C c \oplus C d$ be the extended Cartan subalgebra, and let $h_{0} = c - θ^{\lor}$ be the affine simple coroot. The level-one affine node-zero fundamental weight is the unique linear functional $Λ_{0} \in {\hat{h}}^{*}$ satisfying

$Λ_{0} ∣_{h} = 0, Λ_{0} (c) = 1, Λ_{0} (d) = 0.$

Thus $Λ_{0} (h_{i}) = 0$ for every finite simple coroot $h_{i}$ , and $Λ_{0} (h_{0}) = 1$ . For each positive integer $ℓ$ , the affine weight $ℓ Λ_{0}$ has level $ℓ$ , affine simple-coroot labels $ℓ$ at node $0$ and $0$ at every finite node, and degree-derivation eigenvalue $0$ .

Definition references

Degree derivation and full untwisted affine algebra

Null root, central coroot, and affine level

The affine simple root alpha zero is delta minus the highest root

Depends on

Dependency tree · 2 levels

Sources

Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras