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.

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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

Used by

Casimir operator relative to an invariant form Definition
Killing form Definition
Trace forms are symmetric and invariant Proposition

Dependency tree · two levels

9 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

Kirillov, An Introduction to Lie Groups and Lie Algebras, §6.1 (standard reference, not scraped)