The Levi Decomposition Theorem, in Lie theory, states that every finite-dimensional Lie algebra over a field of characteristic zero can be written as a semidirect product of two pieces: its radical, the maximal solvable ideal contained in the algebra, and a semisimple subalgebra called a Levi subalgebra, complementary to the radical. The result was conjectured by Wilhelm Killing and Elie Cartan and proved by Eugenio Elia Levi, and it reduces much of the study of general finite-dimensional Lie algebras to the two more tractable cases of solvable and semisimple Lie algebras.
Facts
StatementEvery finite-dimensional Lie algebra g over a field of characteristic zero can be written as the semidirect product of a solvable ideal, its radical, and a semisimple subalgebra, a Levi subalgebra. 1 Classification
Statement Form Connections
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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 Levi decomposition, Wikipedia
Proved By
Source Levi decomposition, Wikipedia
Sources
1. Levi decomposition, Wikipedia
Lead section
any finite-dimensional Lie algebra g over a field of characteristic zero is the semidirect product of a solvable ideal and a semisimple subalgebra.
History section
proved by Eugenio Elia Levi (1905)
- In Branch: Representation Theory, Lead sentence
Proved By: Eugenio Elia Levi, Lead paragraph
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi (1905), states that any finite-dimensional
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