Semidirect product of Lie algebras

The semidirect product $𝔞 ⋊ 𝔟$ of Lie algebras is a generalization of the Direct product of Lie algebras where only one of the operands is required to be an ideal.¹. The semidirect product $𝔞 ⋊ 𝔟$ is an extension of $𝔟$ by $𝔞$ ,

0 \to 𝔞 ↪ 𝔞 ⋊ 𝔟 ↠ 𝔟 \to 0

and extensions which can be written this way are precisely split extensions.

Internal semidirect product.

Let $𝔞 ⊴ 𝔤$ and $𝔟 \leq 𝔤$ be subalgebras, the first of which is an ideal, such that $𝔤 = 𝔞 \oplus 𝔟$ internally. Then $𝔤$ is the internal semidirect product $𝔞 ⋊ 𝔟$ .

External semidirect product

Let $𝔞$ be a Lie algebra and let $𝔟$ be a Lie algebra acting on $𝔞$ by derivations, i.e. equipped with a Lie algebra homomorphism $𝜋 : 𝔟 \to 𝔡 𝔢 𝔯 (𝔞)$ into the Derivation subalgebra $𝔡 𝔢 𝔯 (𝔞) \leq E n d 𝔞$ , so that $𝜋 (𝑥)$ is a derivation of $𝔞$ for every $𝑥 \in 𝔟$ . Then the external semidirect product $𝔞 ⋊ 𝔟$ is the unique Lie bracket on the sum vector space $𝔞 \oplus 𝔟$ such that $𝔞$ and $𝔟$ are subalgebras and #m/def/lie

a d 𝑥 𝑦 = [𝑥, 𝑦] = 𝜋 (𝑥) 𝑦

for all $𝑦 \in 𝔞$ and $𝑥 \in 𝔟$ .

Properties

$𝔞 ⋊ 𝔟 ≅ 𝔞 \times 𝔟$ iff $𝜋 = 0$ is the trivial representation

Special cases

Adjoining a derivation

Semidirect product of Lie algebras

Internal semidirect product.

External semidirect product

Properties

Special cases

See also

Footnotes