---- > [!definition] Definition. ([[Lie algebra homomorphism]]) > Let $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ and $(\mathfrak{h}, [-,-]_{\mathfrak{h}})$ be [[Lie algebra|Lie algebras]] over a [[field|field]] $k$. A **Lie algebra homomorphism** is a [[linear map]] $\varphi: \mathfrak{g} \to \mathfrak{h}$ that is compatible with the Lie bracket: $\varphi([x, y]_{\mathfrak{g}})=[\varphi(x), \varphi(y)]_{\mathfrak{h}}$ for all $x,y \in \mathfrak{g}$. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```