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.

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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.

Depends on

Used by

Subrepresentations, quotient representations, and intertwiners Definition
Adjoint and trivial representations Example
Standard representations of classical matrix Lie algebras Example
Direct-sum, dual, Hom, and tensor representations Proposition
Lie representations as actions before enveloping Proposition
Representation kernels are ideals Proposition
Lie representations are U(g)-modules Theorem

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

Etingof, MIT 18.745 notes, §11.1, printed pp. 61–62 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §4.1, printed pp. 49–50 (standard reference, not scraped)