---- > [!theorem] Theorem. ([[the classical Lie algebras under the Cartan-Killing classification]]) > Under the [[classification of complex semisimple Lie algebras|Cartan-Killing classification]], > > The [[Dynkin diagram|Dynkin diagrams]] [[classification of irreducible root systems|corresponding to]] [[simple Lie algebra|simple]] [[Lie algebra|Lie algebras]] are, as discussed in the [[classification of complex semisimple Lie algebras|Cartan-Killing classification]], of type $A_{n},B_{n},C_{n},D_{n}$. It turns out that under the [[classification of complex semisimple Lie algebras|Cartan-Killing classification]] these correspond rather directly to the **classical Lie algebras** $\mathfrak{sl}_{n+1}$, $\mathfrak{so}_{2n+1}$, $\mathfrak{sp}_{2n}$, $\mathfrak{so}_{2n}$ ($n>1$). Specifically, we arrive at the following table. > > > > > > | Type | Ambient Space | Root System $\Phi$ | Simple Roots $\Delta$ | Weyl Group | Lie Algebra $\mathfrak{g}$ | Dimension | > |----------|-----------------------------------------------|-------------------------------------------------------------------|------------------------------------------------------------------------|--------------------------|-----------------------------|-----------------| > | $A_n$ | $\{x \in \mathbb{R}^{n+1} : \sum x_i = 0\}$ | $\{ e_i - e_j : i \ne j \}$ | $\{ e_1 - e_2, \dots, e_n - e_{n+1} \}$ | $S_{n+1}$ | $\mathfrak{sl}_{n+1}$ | $(n+1)^2 - 1$ | > | $B_n$ | $\mathbb{R}^n$ | $\{ \pm e_i \} \cup \{ \pm e_i \pm e_j : i \ne j \}$ | $\{ e_1 - e_2, \dots, e_{n-1} - e_n, e_n \}$ | $S_n \ltimes C_2^n$ | $\mathfrak{so}_{2n+1}$ | $2n^2 + n$ | > | $C_n$ | $\mathbb{R}^n$ | $\{ \pm 2e_i \} \cup \{ \pm e_i \pm e_j : i \ne j \}$ | $\{ e_1 - e_2, \dots, e_{n-1} - e_n, 2e_n \}$ | $S_n \ltimes C_2^n$ | $\mathfrak{sp}_{2n}$ | $2n^2 + n$ | > | $D_n$ | $\mathbb{R}^n$ ($n \geq 4$) | $\{ \pm e_i \pm e_j : i \ne j \}$ | $\{ e_1 - e_2, \dots, e_{n-1} - e_n, e_{n-1} + e_n \}$ | $S_n \ltimes C_2^{n-1}$ | $\mathfrak{so}_{2n}$ | $2n^2 - n$ | > > Mnemonic for $\Delta$: > **A** has **a**ll adjacent differences (and that is all) > **B** has all adjacent differences, but also the baby **b**aby root-length (i.e. has $e_{n}$) > **C** has all adjacent differences, but also the chunky **c**hunky double roots (i.e., has $\{ 2e_{n} \}$) > **D** has all adjacent differences, but also, but diverges (last is $e_{n-1}-e_{n}$ and $e_{n-1}+e_{n}$, representing the split in the Dynkin diagram) > [!proof]- Proof. ([[the classical Lie algebras under the Cartan-Killing classification]]) > ~ > > **$A_{n}$.** Recall that the [[root space decomposition of sln(C)|rsd]] of $\mathfrak{sl}_{n+1}(\mathbb{C})$ is $\mathfrak{sl}_{n+1}(\mathbb{C})= \mathfrak{t} \oplus \bigoplus_{i \neq j} \mathfrak{g}_{e_{i}-e_{j}}$ where $\mathfrak{t}$ is the [[diagonal matrix|diagonal]] [[Cartan subalgebra|CSA]] and $e_{i} \in \mathfrak{t^{*}}$ returns the $ii$th entry of its argument. In other words, $\Phi=(e_{i}-e_{j})_{i \neq j} \subset \mathfrak{t}^{*}$ is the [[root system of a Lie algebra|root system]]. Moreover $\Delta=\{ e_{1}-e_{2},\dots,e_{n}-e_{n+1} \}=\{ \alpha_{1},\dots,\alpha_{n} \}$ is manifestly a [[basis]] of [[root basis|simple roots]], and using (for example) [[root string in a Lie algebra|root strings]] we may compute that if $i \neq j$ then $\langle \alpha_{i}, \check{\alpha}_{j} \rangle = \begin{cases} > -1 & |i-j| = 1 \\ > 0 & \text{else.} > \end{cases}$ > Thus the [[cartan matrix]] of $\Phi$ is the same as that of the $A_{n}$ [[root system]], so $\Phi$ has type $A_{n}$. > > > Others are similar and in handwritten notes. > > > **$\mathfrak{so}_{6}$.** $6=3 \cdot 2$; ---- #### - [[general linear Lie algebra]] - [[special linear Lie subalgebra]] - [[orthogonal Lie algebra]] - [[symplectic Lie algebra]] - [[root basis]] - [[Weyl group of a root system]] ----- #### 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 > ```