DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Chevalley-contravariant forms

Definition

Fix simple roots ${α_{i}}$ in the chosen positive system and normalized Chevalley generators $e_{i} \in g_{α_{i}}$ , $f_{i} \in g_{- α_{i}}$ . Let $τ : U (g) \to U (g)$ be the Chevalley anti-involution determined by $τ (e_{i}) = f_{i}$ , $τ (f_{i}) = e_{i}$ , and $τ (h) = h$ for $h \in h$ . A bilinear form $B$ on a $g$ -module is Chevalley-contravariant when

$B (x u, v) = B (u, τ (x) v) (x \in U (g)) .$

This is a bilinear condition, not a Hermitian or positivity condition.

Depends on

Used by

Dependency tree · two levels

5 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

Mrudul Thatte, Category O: Verma's Thesis, Definition 1.1 (standard reference, not scraped)