---- > [!definition] Definition. ([[root system of a Lie algebra]]) > Let $\mathfrak{g}$ be a [[semisimple Lie algebra|semisimple]] $\mathbb{C}$-[[Lie algebra]], $\mathfrak{t} \subset \mathfrak{g}$ a [[Cartan subalgebra]]. Consider the [[root space decomposition of a Lie algebra|root space decomposition]] $\mathfrak{g}=\mathfrak{t} \oplus \bigoplus_{\alpha \in \Phi}\mathfrak{g}_{\alpha}.$ > [[root space decomposition of a Lie algebra|Recall]] that the [[killing form]] $\kappa$ restricted to $\mathfrak{t}$ defines [[dual vector space|an]] [[isomorphism]] $\mathfrak{t} \to \mathfrak{t^{*}}$, whose inverse we denote by $t_{\lambda} \leftarrow_{{|}} \lambda$. This isomorphism lets us transport the killing from $\mathfrak{t}$ to $\mathfrak{t}^{*}$: if $\lambda, \mu \in \mathfrak{t}^{*}$, define $(\lambda, \mu):= \kappa(t_{\lambda}, t_{\mu}).$ > [[The Cartan-Killing Criterion|Then]] $(-,-):\mathfrak{t}^{*} \times \mathfrak{t}^{*} \to \mathbb{C}$ is a [[symmetric multilinear map|symmetric]] [[nondegenerate bilinear form|nondegenerate]] [[bilinear map|bilinear form]]. Note that $(\lambda, \mu)=\lambda(t_{\mu})=\mu(t_{\lambda})$. > > Let $E=\mathbb{R}\Phi \subset \mathfrak{t}^{*}$ be the $\mathbb{R}$-[[submodule generated by a subset]] of $\Phi$. We can prove the following: > 1. The restriction of $(-,-)$ to $E\times E$ is [[positive definite bilinear form|positive definite]] and $\mathbb{R}$-valued, meaning that $\big(E, (-,-) \big)$ is a Euclidean space. > 2. If $\alpha \in \Phi$ and $c \in \mathbb{R}$, then $c \alpha \in \Phi \iff c=\pm 1$ > 3. If $\alpha, \beta \in \Phi$, then $\beta - \frac{2(\beta, \alpha)}{(\alpha, \alpha)} \alpha \in \Phi$ > 4. If $\alpha, \beta \in \Phi$, then $\frac{2(\beta, \alpha)}{(\alpha , \alpha)} \in \mathbb{Z}$. > > In other words, the triple $\big( \Phi, E, (-,-) \big)$ is a [[root system]]. It turns out that this data (up to a suitable [[isomorphism]]) exactly classifies the [[semisimple Lie algebra]] $\mathfrak{g}$: [[classification of complex semisimple Lie algebras]]. ---- #### ---- #### 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 > ```