Lie algebras MOC

Direct product of Lie algebras

The direct product is the categorical product in Category of Lie algebras. Given two Lie algebras 𝔤,𝔥, their product 𝔤 ×𝔥 consists of the direct product vector space¹ together with a bracket given by

[(𝑔1,ℎ1),(𝑔2,ℎ2)]=([𝑔1,𝑔2],[ℎ1,ℎ2])

Hence regarded as subspaces 𝔤 ≅(𝔤,0) and 𝔥 ≅(0,𝔥) commute and are ideals.

Internal direct product

Let 𝔞,𝔟 ≤𝔤 be subalgebras such that 𝔤 =𝔞 ⊕𝔟 and [𝔞,𝔟] =0. Then 𝔤 is the internal direct product 𝔞 ×𝔟. Equivalently, 𝔤 =𝔞 ⊕𝔟 with both 𝔞,𝔟 ⊴𝔤 ideals.

See also

• Semidirect product of Lie algebras, which generalizes the direct product

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Footnotes

1. for two (or by induction, finite) operands this is naturally isomorphic to the direct sum of vector spaces 𝔤 ⊕𝔥 ↩