---- > [!theorem] Theorem. ([[any representation of sl2(C) is completely reducible]]) > Any [[Lie algebra representation|representation]] of the [[Lie algebra]] $\mathfrak{sl}_{2}(\mathbb{C})$ is [[completely reducible]]. ^theorem > [!proposition] Corollary. > - [[weights characterize any representation of sl2(C)]] ^proposition > [!note] Remark. > A somewhat more elegant proof comes from studying [[group]] [[group representation|representations]] of the [[Lie group]] $\text{SU}(2)$. The upshot of the proof provided here is that it has lots of techniques that we'll see in the future. ^note > [!proof]- Proof. ([[any representation of sl2(C) is completely reducible]]) > ~ $V$ is a [[vector space]]. Recall the usual [[basis]] $\{ e,h, f \}$ [[special linear Lie subalgebra#^basic-example|for]] $\mathfrak{sl}_{2}(\mathbb{C})$. The proof makes use of the following definition: > [!definition] Definition. (Casimir element) > > Let $\rho:\mathfrak{sl}_{2}(\mathbb{C}) \to \mathfrak{gl}(V)$ be a [[Lie algebra representation|representation]]. The **Casimir element of $\rho$** is the [[linear map]] $\Omega_{\rho}=\rho(e) \rho(f)+ \rho(f) \rho(e) + \frac{1}{2} \rho(h)^{2} \in \mathfrak{gl}(V)=\text{End }V.$ It is in fact $\mathfrak{g}$-equivariant, i.e., a [[morphism of Lie algebra representations]] $V \to V$. This is a big deal: equivariant maps are rare! > Later on, we will give a more general definition: [[Casimir element]]. ^definition Begin with two easy lemmas. > [!proposition] Lemma 1. > If $\rho=V(n)$ is the [[irreducible Lie algebra representation|irreducible]] [[Lie algebra representation|representation]] [[classification of the irreps of sl2 over C|of]] $\mathfrak{sl}_{2}(\mathbb{C})$ of [[dimension]] $n+1$, then $\Omega_{\rho}=c \cdot \id$, where $c=n+\frac{n^{2}}{2}$. ^proposition The [[Casimir element]] $\Omega_{\rho}$ is $\mathfrak{sl}_{2}(\mathbb{C})$-equivariant; by [[Schur's Lemma for Lie algebras|Schur's lemma]], $\Omega_{\rho}=c \id_{V}$ for some $c \in \mathbb{C}$. To compute $c$, determine the action of $\Omega_{\rho}$ on a highest weight vector. Computing with the explicit description of $V(n)$ in [[classification of the irreps of sl2 over C]] gives the result. > [!proposition] Lemma 2. > Suppose $\mathfrak{h}$ is an [[abelian Lie algebra]]. Then every [[Lie algebra homomorphism]] $\mathfrak{sl}_{2}(\mathbb{C}) \to \mathfrak{h}$ is zero. ^proposition The following proposition, which is proved via 3 cases, is now the meat of the proof. > [!proposition] > If $W \subset V$ has codimension $1$, then it has a [[complement of a linear subspace|complementary]] [[Lie algebra subrepresentation|subrepresentation]]. In particular, there exists a *[[trivial group representation|trivial]]* [[Lie algebra representation|representation]] $W' \subset V$ such that $V=W \oplus W'$. ^proposition Finally, we are ready to show complete reducibility. Using [[Lie algebra representation completely reducible iff every subrepresentation admits a complement]], it is enough to show that any [[Lie algebra subrepresentation|subrepresentation]] $W$ of $V$ has a complement in $V$. (Bring over last part from handwritten notes) ---- #### ----- #### 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 > ```