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, #m/def/lie i.e. a Lie algebra on (additive) cosets of $𝔞$ with the well-defined product

[𝑥 + 𝔞, 𝑦 + 𝔞] = [𝑥, 𝑦] + 𝔞

for $𝑥, 𝑦 \in 𝔤$ and the canonical projection

𝜋 : 𝔤 \to 𝔤 / 𝔞 𝑥 \mapsto 𝑥 + 𝔞

yielding the short exact sequence

0 \to 𝔞 𝜄 ↪ 𝔤 𝜋 ↠ 𝔤 / 𝔞 \to 0

#state/tidy | #lang/en | #SemBr