----- >[!proposition] Proposition. ([[Schur's Lemma for Lie algebras]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] over a [[field]] $\mathbb{F}$. Let $V,W$ be [[irreducible Lie algebra representation|irreducible]] $\mathfrak{g}$-[[Lie algebra representation|representations]] and $\varphi:V\to W$ a $\mathfrak{g}$-[[morphism of Lie algebra representations|homomorphism]]. Then: > > 1. Either $\varphi=0$ or $\varphi$ is an [[isomorphism]]; > 2. If $\mathbb{F}$ is [[algebraically closed]] (e.g. $\mathbb{C}$) and $V=W$ then in particular $\varphi=\lambda I$ for some scalar $\lambda \in \mathbb{F}$ ^proposition > [!proof]- Proof. ([[Schur's Lemma for Lie algebras]]) > **1.** $\operatorname{ker }\varphi$ is a [[Lie algebra subrepresentation|subrepresentation]] of $V$: since $V$ is irreducible, either $\operatorname{ker }\varphi=V$ (so $\varphi=0$) or $\operatorname{ker }\varphi=(0)$ (so $\varphi$ is [[injection|injective]]). By considering the image of $\varphi$ similarly, we see $\varphi$ is either zero or an isomorphism. > > **2.** Since $\mathbb{F}$ is [[algebraically closed]], $\varphi$ has an [[eigenvalue]] $\lambda$. By irreducibility, the $\lambda$-[[eigenspace]] must be all of $V$, i.e., $\varphi v=\lambda v$ for all $v \in V$. This is saying precisely that $\varphi=\lambda \ \id$. ----- #### ---- #### 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 > ```