Lie algebra extension

Let $𝔞, 𝔟$ be Lie algebras. An extension of $𝔞$ by $𝔟$ is a Lie algebra $𝔤$ together with a short exact sequence lie

0 \to 𝔟 ↪ 𝔤 ↠ 𝔞 \to 0

Hence the $𝔤$ “covers” $𝔞$ with kernel $𝔟 ↪ 𝔤$ . Note that $𝔟$ is necessarily an ideal, giving the quotient $𝔤 / 𝔟 ≅ 𝔞$ by the First isomorphism theorem. Two extensions $𝔤, 𝔤 1$ of $𝔞$ by $𝔟$ are said to be equivalent iff there exists an isomorphism such that the following diagram commutes

Classification

Consider an extension $0 \to 𝔟 ↪ 𝔤 𝑝 ↠ 𝔞 \to 0$ .

Iff $𝔟$ is abelian, one speaks of an abelian extension,
Iff $𝔟 ↪ 𝔤$ is a central ideal, one speaks of a central extension.
Iff $𝔤 ≅ 𝔟 ⋊ 𝔞$ (Semidirect product of Lie algebras), one speaks of a split extension, equivalently $𝑝$ is split epic.
Iff $𝔤 ≅ 𝔟 \times 𝔞$ (Direct product of Lie algebras), one speaks of a trivial extension.

Proof of equivalence in 3.

proof

Table of Contents

Lie algebra extension

Classification

See also

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