Weyl's Theorem on Complete Reducibility is a fundamental result in the representation theory of Lie algebras, named for Hermann Weyl. It states that for a semisimple Lie algebra over a field of characteristic zero, every finite-dimensional representation, or module, of that algebra decomposes as a direct sum of irreducible representations, so any such representation is completely reducible with no indecomposable pieces left over beyond the irreducibles themselves. The theorem is a cornerstone of the classical theory of Lie algebra representations, underlying the classification of finite-dimensional representations of semisimple Lie algebras.
Facts
StatementFor a semisimple Lie algebra g over a field of characteristic zero, every finite-dimensional module over g is semisimple as a module, that is, a direct sum of simple modules. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Weyl's theorem on complete reducibility, Wikipedia
Proved By
Source Weyl's theorem on complete reducibility, Wikipedia
Sources
1. Weyl's theorem on complete reducibility, Wikipedia
Statement section
every finite-dimensional module over g is semisimple as a module (i.e., a direct sum of simple modules.)
In Branch: Representation Theory, Lead sentence
Lie algebra representations (specifically in the representation theory of semisimple Lie algebras).
Proved By: Hermann Weyl, Lead paragraph
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in
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