[[Lie algebras MOC]]
# Lie algebra homomorphism

A **Lie algebra homomorphism** is an [[Algebra homomorphism]] of [[Lie algebra|Lie algebras]] over the same field $\mathbb{K}$ #m/def/lie
i.e. $f :\mathfrak{g} \to \mathfrak{h}$ such that for $x,y \in \mathfrak{g}$ and $\lambda,\mu \in \mathbb{K}$
1. $f(\lambda x + \mu y)= \lambda f(x) + \mu f(y)$ ^H1
2. $f([x,y])=[f(x),f(y)]$ ^H2

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