---- > [!theorem] Theorem. ([[classification of complex semisimple Lie algebras]]) > From [[root system of a Lie algebra]] and [[classification of irreducible root systems]] we obtain maps between (isomorphism classes of) the following objects: > > $\{ \begin{aligned} > & \ \ \ \text{Pairs } (\mathfrak{g}, \mathfrak{t}), \text{ where} \\ > &\mathfrak{g} \text{ semisimple, } \mathfrak{t} \text{ a CSA} > \end{aligned} \} \to \{ \text{Root systems} \} \leftrightarrow \{ \text{Dynkin diagrams} \}.$ > > > > We now show: > 1. **(Conjugacy Theorem)** The first map is [[well-defined]] independent of the choice of [[Cartan subalgebra]] $\mathfrak{t}$. As corollaries, have > 1. If $\Phi, \Phi'$ [[root system|root systems]] of $\mathfrak{g}$ in $\mathfrak{t}^{*}$, $\mathfrak{t}'^{*}$, then $\Phi \cong \Phi'$ > 2. We have $\mathfrak{g} \text{ simple } \iff \Phi \text{ irred.} \iff \text{D}(\Phi) \text{ is connected}$ > 2. **(Cartan-Killing Classification)** The first map is a [[bijection|bijection]]. This comes from the following very strong statement on existence and uniqueness. Let $\mathfrak{g,\mathfrak{g'}}$ be [[semisimple Lie algebra|semisimple]] [[Lie algebra|Lie algebras]] with [[Cartan subalgebra|CSAs]] $\mathfrak{t}, \mathfrak{t}'$ respectively, giving [[root system|root systems]] $\Phi,\Phi'$. Let $e_{\alpha} \in \mathfrak{g}_{\alpha}$, $e_{\alpha'} \in \mathfrak{g}_{\alpha'}$ be choices of nonzero elements for each $\alpha \in \Delta$ and $\alpha' \in \Delta'$. Let $f:\Phi \to \Phi'$ be a [[root system isomorphism]] sending $\Delta \mapsto \Delta'$. *Then* there is a unique [[Lie algebra homomorphism|isomorphism of Lie algebras]] $\tilde{f}:\mathfrak{g} \to \mathfrak{g}'$ with the following property: $\mathfrak{t} \xmapsto{f} \mathfrak{t}'$, > > Moreover, the bijection restricts to a bijection $\{\text{\textit{simple} }\mathfrak{g} \} \leftrightarrow \text{irred. root systems} \leftrightarrow \text{connected Dynkin diagrams}.$ > (This part is not hard to prove.) > > This is enough content to substantiate a note of its own. It is of additional importance to understand the Cartan-Killing classification explicitly for the classical Lie algebras. See: [[the classical Lie algebras under the Cartan-Killing classification]]. ---- #### - [[semisimple Lie algebra|semisimple]] - [[root system|root]] - [[Dynkin diagram]] ----- #### 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 > ```