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.

Representations of Lie algebras

Definition

For a vector space $V$ over $k$ , write

$g l (V) = {End}_{k} (V)$

with commutator bracket $[A, B] = A B - B A$ . Expansion of triple composites shows that this is a Lie algebra in every characteristic.

A representation of a Lie algebra $g$ on $V$ is a Lie-algebra homomorphism

$ρ : g ⟶ g l (V) .$

Writing $x v$ for $ρ (x) (v)$ , this is equivalently a $k$ -bilinear action $g \times V \to V$ satisfying

$[x, y] v = x (y v) - y (x v)$

for all $x, y \in g$ and $v \in V$ . No finite-dimensional hypothesis is imposed on either $g$ or $V$ . The zero vector space and the zero action are allowed.

Representations of Lie algebras

Definition

Depends on

Dependency tree · 2 levels

Sources