DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Trace form of a representation

Definition

Let $ρ : g \to g l (V)$ be a representation for which $V$ is finite-dimensional. Its trace form is the scalar-valued function

$B_{ρ} (x, y) = tr (ρ (x) ρ (y)) (x, y \in g) .$

The trace is the basis-independent trace from The basis-independent trace of an endomorphism of a finite-dimensional vector space, so the definition does not depend on a basis of $V$ . The acting algebra $g$ need not be finite-dimensional. The finite-dimensionality of $V$ is essential here because it is the standing hypothesis under which the cited trace has been defined. No nondegeneracy, faithfulness, or symmetry is asserted in the definition.

Depends on

Dependency tree · 2 levels

Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, §6.1