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

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Semidirect products of Lie algebras

Definition

By Derivations form a Lie algebra and inner derivations an ideal, $Der (h)$ is a Lie algebra under the commutator. Let $ρ : g \to Der (h)$ be a Lie-algebra homomorphism. On $g \oplus h$ define the candidate bracket

$[(x, u), (y, v)] = ([x, y], ρ (x) v - ρ (y) u + [u, v]) .$

The next lemma proves that this bracket satisfies Jacobi. The resulting Lie algebra is the semidirect product $g ⋉_{ρ} h$ . The subspace $0 \oplus h$ is an ideal, the projection to $g$ is a Lie homomorphism, and the inclusion of $g$ is a section. When $ρ = 0$ , this is the direct sum Lie algebra.

Depends on

Dependency tree · 2 levels

Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, semidirect-product convention in §3.3