----- > [!proposition] Proposition. ([[finding sl2-triples]]) > Let $\mathfrak{g}$ be a [[semisimple Lie algebra|semisimple]] [[Lie algebra]] over $\mathbb{C}$ with [[root space decomposition of a Lie algebra|root space decomposition]] $\mathfrak{g}=\mathfrak{t} \oplus \bigoplus_{\alpha \in \Phi}\mathfrak{g}_{\alpha}.$ We use notation from the [[root space decomposition of a Lie algebra|rsd]] note. > Fixing some $\alpha \in \Phi$, define $h_{\alpha}:= \frac{2t_{\alpha}}{\kappa(t_{\alpha}, t_{\alpha})} \in \mathfrak{t}$ which is safe to do because [[root space decomposition of a Lie algebra|we've shown]] $\kappa(t_{\alpha},t_{\alpha})\neq0$. > Then we have for any nonzero element $e_{\alpha} \in \mathfrak{g}_{\alpha}$ that there exists some $f_{\alpha} \in \mathfrak{g}_{-\alpha}$ such that $(e_{\alpha}, h_{\alpha}, f_{\alpha})$ is an **$\mathfrak{sl}_{2}(\mathbb{C})$-triple**, in the sense that it satisfies the [[special linear Lie subalgebra|relations]] $[h_{\alpha}, e_{\alpha}]=2e_{\alpha}, \ [h_{\alpha}, f_{\alpha}]=-2f_{\alpha}, [e_{\alpha}, f_{\alpha}]=h_{\alpha}.$ We shall denote by $\mathfrak{m}_{\alpha}:=\text{span}(e_{\alpha}, h_{\alpha}, f_{\alpha}) \cong \mathfrak{sl}_{2}(\mathbb{C}).$ the [[Lie subalgebra|subalgebra]] of $\mathfrak{g}$ spanned by this triple. > Thus, every root gives rise to a copy of $\mathfrak{sl}_2(\mathbb{C})$ inside $\mathfrak{g}$. > [!note] Remark. > Note that $h_{\alpha}$ doesn't involve choices, but $e_{\alpha}$ (and in turn $f_{\alpha}$) do. However, as a corollary of [[root spaces are one-dimensional]], one has that $\mathfrak{m}_{\alpha}=\mathfrak{g}_{\alpha} \oplus [\mathfrak{g}_{\alpha}, \mathfrak{g}_{-\alpha}]\oplus \mathfrak{g}_{-\alpha},$ and so $\mathfrak{m}_{\alpha}$ is determined independent of any choices. ^note > [!proof]- Proof. ([[finding sl2-triples]]) > Recall that if $\alpha, \beta \in \mathfrak{t}^{*}$ and $\alpha+ \beta \neq 0$, $\mathfrak{g}_{\alpha} \perp \mathfrak{g}_{\beta}$. [[The Cartan-Killing Criterion|Recall]] [[nondegenerate bilinear form|nondegeneracy]] of $\kappa(-,-)$. These mean it can't be the case that $e_{\alpha}$ is orthogonal to every element of $\mathfrak{g}_{-\alpha}$: that would make $\kappa(e_{\alpha}, -)$ the zero map, contradicting nondegeneracy. Thus, there exists $f_{\alpha} \in \mathfrak{g}_{-\alpha}$ such that $\kappa(e_{\alpha}, f_{\alpha}) \neq 0$. Let's *choose* it to be scaled so that $\kappa(e_{\alpha}, f_{\alpha})=\frac{2}{\kappa(t_{\alpha}, t_{\alpha})}.$[^1] Now by [[root space decomposition of a Lie algebra|Property 3.1]]: $\kappa(e_{\alpha }, f_{\alpha})t_{\alpha}=[e_{\alpha}, f_{\alpha}]$, so $\frac{2t_{\alpha}}{\kappa(t_{\alpha}, t_{\alpha})}={[e_{\alpha}, f_{\alpha}]}.$ > This gives $[e_{\alpha}, f_{\alpha}]=\frac{2t_{\alpha}}{\kappa(t_{\alpha, t_{\alpha}})}=h_{\alpha}$. > > Moreover, since $h_{\alpha} \in \mathfrak{t}$, $\begin{align} > [h_{\alpha}, e_{\alpha}]&= \alpha(h_{\alpha}) e_{\alpha} \\ > &= 2 \frac{\cancel{ \alpha( t _{\alpha}) }^{\kappa(t_{\alpha}, t_{\alpha})}}{\kappa(t_{\alpha}, t_{\alpha})} e_{\alpha} \\ > &= 2 e_{\alpha}. > \end{align}$ > Similar computation for $f_{\alpha}$. > > (why is it made to sound like [[Lie's Theorem]] is relevant here? I guess it is used to prove Property 3.3) ----- #### [^1]: The motivation is that we want $[e_{\alpha}, f_{\alpha}]=h_{\alpha}$. Recall (Property 3.1) since $e_{\alpha} \in \mathfrak{g}_{\alpha}$ and $f_{\alpha} \in \mathfrak{g}_{-\alpha}$, $[e_{\alpha}, f_{\alpha}]=\kappa(e_{\alpha}, f_{\alpha})h_{\alpha}$. So we really want $\kappa(e_{\alpha}, f_{\alpha})t_{\alpha}=h_{\alpha}$. Applying $\alpha(\cdot)$ on both sides gives the equality $\kappa(e_{\alpha, }f_{\alpha}) \kappa(t_{\alpha}, t_{\alpha})=2$. Thus, we should choose $f_{\alpha}$ to make this equality true, i.e., choose so that $\kappa(e_{\alpha}, f_{\alpha})=\frac{2}{\kappa(t_{\alpha}, t_{\alpha})}$ . ---- #### 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 > ```