---- > [!definition] Definition. ([[faithful Lie algebra representation]]) > Let $\mathfrak{g}$ be a [[Lie algebra]], $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ a [[Lie algebra representation]]. We call $(\rho ,V)$ **faithful** if the [[Lie algebra homomorphism]] $\rho$ is [[injection|injective]]. > Equivalently, $(\rho, V)$ is faithful iff $\operatorname{ker }\rho=\{ 0 \}$, that is, iff $x \cdot v = 0 \ \fa v \implies x=0.$ ^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 > ```