Quotient Lie algebra

Let $\mathfrak g$ be a Lie algebra (see Lie algebra) and let $\mathfrak i\subseteq \mathfrak g$ be an ideal (see ideal). The quotient Lie algebra $\mathfrak g/\mathfrak i$ is the vector space quotient equipped with the bracket

This is well-defined precisely because $\mathfrak i$ is an ideal: changing representatives adds elements of $\mathfrak i$, and brackets with elements of $\mathfrak i$ stay in $\mathfrak i$.

The projection map $\pi:\mathfrak g\to \mathfrak g/\mathfrak i$ is a Lie algebra homomorphism (see Lie algebra homomorphism) with kernel $\mathfrak i$. It satisfies the universal property: any Lie algebra homomorphism $f:\mathfrak g\to \mathfrak h$ with $\mathfrak i\subseteq \ker(f)$ factors uniquely through $\pi$.

Quotients appear constantly in structure theory. For example, the derived subalgebra $[\mathfrak g,\mathfrak g]$ is an ideal (see derived subalgebra is an ideal), so the abelianization $\mathfrak g/[\mathfrak g,\mathfrak g]$ is a quotient Lie algebra. On the group side, quotients by normal subgroups (see quotient Lie group) differentiate to quotient Lie algebras under mild hypotheses (see differential is a Lie algebra homomorphism).