Lie algebras MOC

Quotient Lie algebra

Given a Lie algebra 𝔤 and a Lie algebra ideal 𝔞 ⊴𝔤, the quotient Lie algebra 𝔤/𝔞 is the quotient algebra, lie i.e. a Lie algebra on (additive) cosets of 𝔞 with the well-defined product

[𝑥+𝔞,𝑦+𝔞]=[𝑥,𝑦]+𝔞

for 𝑥,𝑦 ∈𝔤 and the canonical projection

𝜋:𝔤→𝔤/𝔞𝑥↦𝑥+𝔞

yielding the short exact sequence

0→𝔞𝜄↪𝔤𝜋↠𝔤/𝔞→0

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