---- > [!definition] Definition. ([[Lie algebra subrepresentation]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] and $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ a [[Lie algebra representation|representation]] of $\mathfrak{g}$. We call a [[linear subspace]] $W \subset V$ a **subrepresentation** of $(\rho, V)$ if the $\mathfrak{g}$-action restricts to it, that is, if $x \cdot w \in W \text{ for all } x \in \mathfrak{g}, w \in W.$ ^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 > ```