DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Levi subalgebras and Levi decompositions

Definition

Let $g$ be finite-dimensional with radical $r = rad (g)$ . A Levi subalgebra or Levi factor is a semisimple Lie subalgebra $s \subseteq g$ such that

$g = r \oplus s$

as vector spaces. Since $r$ is an ideal, its bracket action by $s$ makes the map $s ⋉ r \to g$ , $(x, u) \mapsto x + u$ , an isomorphism, where the acting algebra is written first as in Semidirect products of Lie algebras. This is called a Levi decomposition. This definition asserts neither existence nor uniqueness. If $g$ is semisimple, then $r = 0$ and $s = g$ is the evident Levi factor; if $g$ is solvable, a Levi factor, when it exists, must be zero.

Depends on

Used by

A Levi decomposition of the Euclidean-motion algebra of R³ Example
Levi decomposition theorem Theorem

Dependency tree · two levels

11 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

Milne, Lie Algebras, Definition 6.24 (standard reference, not scraped)