---- > [!definition] Definition. ([[morphism of Lie algebra representations]]) > Let $\mathfrak{g}$ be a [[Lie algebra]]. Let $V,W$ be $\mathfrak{g}$-[[Lie algebra representation|representations]]. A **morphism of $\mathfrak{g}$-representations** is a $\mathfrak{g}$-equivariant [[linear map]] $\varphi:V \to W$: for all $x \in \mathfrak{g}$, $v \in V$, $\varphi(x \cdot v)=x \cdot \varphi(v).$ If $\varphi$ is furthermore [[bijection|bijective]] we call it an **isomorphism of $\mathfrak{g}$-representations**. ^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 > ``` > [!metadata]- Created::[[2024-10-15]] Modified::[[2024-10-15]] [[Tags]]:: #definition \ Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Properties:: *[[Properties]]* Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- Let $V,W$ be [[Lie algebra representation|representations]] of a [[Lie algebra]] $\mathfrak{g}$. > [!definition] Definition. ([[representation homomorphism]]) > A [[linear map]] $\varphi:V \to W$ **homomorphism of [[Lie algebra representation|Lie algebra representations]]** is a $\mathfrak{g}$-equivariant map: $\varphi(x \cdot v)=x \cdot \varphi(v)$. > > ^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 > ```