[[Quadratic Lie algebra]]
# Category of quadratic Lie algebras

The **category of quadratic Lie algebras** $\cat{QLie}_{\mathbb K}$ is a [[category]] where
an object is a [[Quadratic Lie algebra]] over $\mathbb K$
and a morphism is a isometric [[Lie algebra homomorphism]]. #m/def/lie 
Thus if $f \in \cat{QLie}_{\mathbb{K}}(\mathfrak{g},\mathfrak{h})$ then
$$
\begin{align*}
\langle f(x),f(y) \rangle_{\mathfrak{h}} = \langle x,y \rangle_{\mathfrak{g}}
\end{align*}
$$
for any $x,y \in \mathfrak{g}$.

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