[[Lie algebra isomorphism]]
# Lie algebra isomorphism theorems

The [[isomorphism theorems]] for [[Lie algebra|Lie algebras]] are expressed as follows.

## First isomorphism theorem

Let $\varphi : \mathfrak{g} \to \mathfrak{h}$ be a [[Lie algebra homomorphism]]. Then the [[quotient Lie algebra|quotient]] by the [[Kernel of a Lie algebra homomorphism|kernel]] is isomorphic to the image: #m/thm/lie 
$$
\begin{align*}
\frac{\mathfrak{g}}{\ker \varphi} \cong \im \varphi \leq \mathfrak{h}
\end{align*}
$$

## Second isomorphism theorem

Let $\mathfrak{a}, \mathfrak{b} \trianglelefteq \mathfrak{g}$. Then #m/thm/lie
$$
\begin{align*}
\frac{\mathfrak{a}+\mathfrak{b}}{\mathfrak{b}} \cong \frac{\mathfrak{a}}{\mathfrak{a} \cap \mathfrak{b}}
\end{align*}
$$

## Third isomorphism theorem

Let $\mathfrak{b} \trianglelefteq \mathfrak{a} \trianglelefteq \mathfrak{g}$ be nested [[Lie algebra ideal|ideals]].
Then $\mathfrak{a} / \mathfrak{b} \trianglelefteq \mathfrak{g} / \mathfrak{a}$ and #m/thm/lie
$$
\begin{align*}
\frac{\mathfrak{g} / \mathfrak{a}}{\mathfrak{a} / \mathfrak{ b}} \cong \frac{\mathfrak{g}}{\mathfrak{b}}
\end{align*}
$$

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