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.

Irreducible, completely reducible, and faithful representations

Definition

A nonzero representation $V$ is irreducible if its only stable linear subspaces are $0$ and $V$ .

A representation is completely reducible if it is isomorphic, as a representation, to an algebraic direct sum

$⨁_{i \in I} V_{i}$

of irreducible representations, where elements have finite support. This is a definition, not an assertion that every representation has such a decomposition. The zero representation is the empty direct sum and is therefore completely reducible under this convention.

A representation $ρ : g \to g l (V)$ is faithful if $ρ$ is injective.

Depends on

Used by

Every finite-dimensional module is a direct sum of highest-weight modules Corollary
Irreducible representations of solvable complex Lie algebras are one-dimensional Corollary
Schur’s lemma for irreducible Lie-algebra representations Corollary
A two-dimensional irreducible real representation of an abelian Lie algebra Counterexample
Positive-characteristic failure of Lie's theorem Counterexample
A nonsplit two-dimensional representation Example
All irreducible finite-dimensional sl2 modules Example
Clebsch–Gordan decomposition for sl2 Example
Exterior powers and fundamental weights of slₙ Example
Standard and dual representations of slₙ Example
Symmetric powers as highest-weight modules Example
A tensor-product top weight does not determine all constituents False statement
Every irreducible real representation of a solvable Lie algebra is one-dimensional False statement
Finite-dimensionality requires dominance integrality False statement
Lie-algebra representations need not be completely reducible False statement
Every finite-dimensional irreducible module has a highest-weight vector Lemma
Unique simple quotient of the dominant cyclic module Lemma
An irreducible module is generated by its highest-weight vector Proposition
Highest weight of the dual representation Proposition
Representation kernels are ideals Proposition
Dominant simple highest-weight modules are finite-dimensional Theorem
Equivalent characterizations of reductive Lie algebras Theorem
Finite-dimensional representations of sl₂ Theorem
Highest-weight classification Theorem
Schur-Weyl decomposition and highest weights Theorem
Simple highest-weight modules are classified by highest weight Theorem
Weyl's complete reducibility theorem Theorem

Dependency tree · two levels

6 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, §4.3, printed pp. 52–53 (standard reference, not scraped)