Lie algebras MOC

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→𝔞↪𝔞⋊𝔟↠𝔟→0

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

Internal semidirect product.

Let 𝔞 ⊴𝔤 and 𝔟 ≤𝔤 be subalgebras, the first of which is an ideal, such that 𝔤 =𝔞 ⊕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 →𝔡𝔢𝔯⁡(𝔞) into the Derivation subalgebra 𝔡𝔢𝔯⁡(𝔞) ≤End⁡𝔞, so that 𝜋(𝑥) is a derivation of 𝔞 for every 𝑥 ∈𝔟. Then the external semidirect product 𝔞 ⋊𝔟 is the unique Lie bracket on the sum vector space 𝔞 ⊕𝔟 such that 𝔞 and 𝔟 are subalgebras and lie

ad𝑥⁡𝑦=[𝑥,𝑦]=𝜋(𝑥)𝑦

for all 𝑦 ∈𝔞 and 𝑥 ∈𝔟.

Properties

• 𝔞 ⋊𝔟 ≅𝔞 ×𝔟 iff 𝜋 =0 is the trivial representation

Special cases

• Adjoining a derivation

See also

• Semidirect product

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Footnotes

1. 1988. Vertex operator algebras and the Monster, p. 7 ↩