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.

Subrepresentations, quotient representations, and intertwiners

Definition

Let $V$ be a representation of $g$ .

A linear subspace $W \subseteq V$ is stable if $x w \in W$ for every $x \in g$ and $w \in W$ . With the restricted action, such a $W$ is a subrepresentation.

For a stable subspace $W$ , the quotient representation on $V / W$ is

$x (v + W) = x v + W .$

This is independent of the representative: replacing $v$ by $v + w$ changes $x v$ by $x w \in W$ . The representation identity descends because both sides are the corresponding cosets of the identity in $V$ .

For representations $V, V^{'}$ of the same Lie algebra, an intertwiner is a linear map $T : V \to V^{'}$ such that

$T (x v) = x T (v)$

for all $x \in g$ and $v \in V$ .

Depends on

Used by

Stability under one generator is not enough Counterexample
Irreducible, completely reducible, and faithful representations Definition
Lie representations are U(g)-modules Theorem

Dependency tree · two levels

8 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)