---- > [!definition] Definition. ([[hom representation]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] over [[field]] $\mathbb{F}$ and suppose $V,W$ are $\mathfrak{g}$-[[Lie algebra representation|representations]]. Then the [[vector space of linear maps between two vector spaces|hom space]] $\text{Hom}_{\mathbb{F}}(V,W)$ is a $\mathfrak{g}$-representation via the action $(x \cdot \varphi)(v) := x \cdot \varphi(v) - \varphi(x \cdot v).$ Note that if $\varphi$ is a [[morphism of Lie algebra representations]] then $x$ acts trivially (i.e., kills) it. 'Quantifies distance from equivariance.' ^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 > ```