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 $v \in V$ .

Subrepresentations, quotient representations, and intertwiners

Definition

Depends on

Dependency tree · 2 levels

Sources