[[Lie algebras MOC]]
# Quotient Lie algebra

Given a [[Lie algebra]] $\mathfrak{g}$ and a [[Lie algebra ideal]] $\mathfrak{a} \trianglelefteq \mathfrak{ g}$, the **quotient Lie algebra** $\mathfrak{g / \mathfrak{ a}}$ is the [[quotient algebra]], #m/def/lie 
i.e. a Lie algebra on (additive) cosets of $\mathfrak{a}$ with the well-defined product
$$
\begin{align*}
[x + \mathfrak{a}, y + \mathfrak{a}] = [x,y] + \mathfrak{a}
\end{align*}
$$
for $x,y \in \mathfrak{ g}$ and the canonical projection
$$
\begin{align*}
\pi : \mathfrak{g} &\to \mathfrak{g}/\mathfrak{a} \\
x &\mapsto x + \mathfrak{a}
\end{align*}
$$
yielding the [[short exact sequence]]
$$
\begin{align*}
0 \to \mathfrak{a} \stackrel{\iota}{\hookrightarrow} \mathfrak{g} \stackrel{\pi}{\twoheadrightarrow} \mathfrak{g}/\mathfrak{a} \to 0
\end{align*}
$$

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