Weyl's Theorem on Complete Reducibility¶
Every finite-dimensional module over a finite-dimensional semisimple Lie algebra in characteristic zero splits into irreducible submodules.
Core Idea¶
Weyl's theorem on complete reducibility states that every finite-dimensional module for a finite-dimensional semisimple Lie algebra over a characteristic-zero field is a direct sum of irreducible modules. Equivalently, every invariant submodule has an invariant complement. The complement must preserve the algebra action; an arbitrary vector-space complement is insufficient. Etingof states this result as Theorem 18.9.[^ref-dac65a04217e]
Scope of Application¶
The guarantee applies to representations with all three hypotheses: semisimple finite-dimensional acting Lie algebra, characteristic-zero field, and finite-dimensional module. It concerns modules over the algebra, rather than a decomposition of the acting algebra itself. A module outside these hypotheses might still split, but that would not follow from this theorem. No uniqueness of the complement is promised.[^ref-dac65a04217e]
Clarity¶
“Completely reducible” means that invariant pieces can be combined as a direct sum. The invariant-complement formulation tests the assertion on an arbitrary invariant part, while the irreducible-summand formulation describes the whole module. Both are stronger than choosing a basis that merely splits the underlying vector space.[^ref-dac65a04217e]
Manages Complexity¶
Once the stated hypotheses are verified, the theorem supplies a general existence guarantee instead of requiring a separate splitting proof for each eligible module. Calculating the particular summands still takes work; the theorem does not itself supply a canonical basis or an algorithm for every representation.[^ref-dac65a04217e]
Abstract Reasoning¶
For an invariant \(U\subseteq V\), Etingof's extension argument splits \(0\to U\to V\to V/U\to0\), yielding an invariant \(W\) with \(V=U\oplus W\). Repetition in finite dimension yields irreducible summands. If the algebra, field, or dimension conditions change, the automatic inference must be reconsidered.[^ref-dac65a04217e]
Knowledge Transfer¶
The same theorem applies to different eligible Lie algebras and modules, including the \(\mathfrak{sl}_{2}\) tensor square and the \(\mathfrak{sl}_{3}\) endomorphism action below. General part/whole reasoning transfers through the Decomposition Prime; this named Lie theorem does not become a cross-domain theorem merely because other subjects also decompose objects.[^ref-dac65a04217e]
Example¶
\(\mathfrak{sl}_{2}\) tensor square (editorial calculation): over characteristic-zero \(k\), the swap of factors commutes with the diagonal \(\mathfrak{sl}_{2}(k)\) action on \(k^2\otimes k^2\). Its symmetric and antisymmetric invariant pieces give \(\operatorname{Sym}^2(k^2)\oplus\Lambda^2(k^2)\), dimensions \(3+1\). Both pieces are irreducible. The acting algebra, whole module, selected invariant part, invariant complement, and irreducible result instantiate the theorem's five roles.[^ref-dac65a04217e]
\(\mathfrak{sl}_{3}\) endomorphism action (editorial calculation): under the commutator action, scalar matrices form an invariant part of \(\operatorname{End}(k^3)\), while trace-zero matrices form its invariant complement. Thus \(\operatorname{End}(k^3)=kI\oplus\mathfrak{sl}_{3}(k)\), dimensions \(1+8\). The latter is the irreducible adjoint module. This maps the same five roles to a different carrier. These two calculations illustrate the theorem; they are not presented as printed examples in Etingof's Theorem 18.9.[^ref-dac65a04217e]
Relationships to Other Abstractions¶
Current abstraction Weyl's Theorem on Complete Reducibility Domain-specific
Parents (2) — more general patterns this builds on
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Weyl's Theorem on Complete Reducibility is a kind of Formal theorem Domain-specific
Weyl complete reducibility is a proven statement with fixed Lie-algebra hypotheses and a splitting conclusion.
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Weyl's Theorem on Complete Reducibility presupposes Decomposition Prime
The theorem's direct-sum conclusion requires decomposition into recombinable invariant summands.
Hierarchy paths (3) — routes to 3 parentless roots
- Weyl's Theorem on Complete Reducibility → Formal theorem → Formal System → Formalization → Representation → Abstraction
- Weyl's Theorem on Complete Reducibility → Decomposition
- Weyl's Theorem on Complete Reducibility → Formal theorem → Formal System → Formalization → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Weyl's Theorem on Complete Reducibility sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Étale Algebra — 0.84
- Fusion Category — 0.83
- Cohomological dimension — 0.81
- Associative algebra — 0.81
- Lie Algebra Extension — 0.81
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
Maschke's theorem treats finite groups, and Peter–Weyl theory treats compact groups under different hypotheses. Weyl law and Weyl algebra share a name but not this assertion. The approved Formal Theorem parent is a strict genus of proven statements; Decomposition is a presupposed whole/parts structure, not the theorem's claim that every decomposition in every subject is guaranteed.[^ref-dac65a04217e]
References¶
[^ref-dac65a04217e]: Pavel Etingof, Lie Groups and Lie Algebras, MIT OpenCourseWare 18.755 lecture notes (2024), §11.4, §17.3, §18.4, Theorem 18.9, p. 100. The two tensor-square and endomorphism examples above are editorial derivations from the cited theory, not examples printed with Theorem 18.9.