----- > [!proposition] Proposition. ([[root space decomposition of sln(C)]]) > For $\mathfrak{g}=\mathfrak{sl}_{n}(\mathbb{C})$ and the [[Cartan subalgebra]] $\mathfrak{t}$ of [[diagonal matrix|diagonal matrices]], define $e_{i} \in \mathfrak{t}^{*}$ by $e_{i} (\begin{bmatrix} > t_{1} & & \\ > & \ddots & \\ > & & t_{n} > \end{bmatrix}) = t_{i}.$ > Then the [[root space decomposition of a Lie algebra|root space decomposition]] of $\mathfrak{g}=\mathfrak{sl}_{n}(\mathbb{C})$ is given by $\mathfrak{g}=\mathfrak{t} \oplus \bigoplus_{i \neq j} \mathfrak{g}_{e_{i}- e_{j}}$ > where $e_{i}-e_{j} \in \mathfrak{t}^{*}$ are the [[simultaneously diagonalizable|simultaneous eigenvalues]] corresponding to simultaneous [[eigenvector|eigenvectors]] $E_{ij}$ (in other words, the standard basis of the [[special linear Lie subalgebra]] is a [[simultaneously diagonalizable|simultaneous]] [[eigenbasis]] for the adjoint action of $\mathfrak{t}$). >[!proof]- Proof. ([[root space decomposition of sln(C)]]) > > Examine [[adjoint representation|adjoint action]] of $\mathfrak{t}$. Given $t \in \mathfrak{t}$, compute $\begin{align} > t \cdot E_{ij} &= [t, E_{ij}] \\ > &= t_{i} E_{ij} - t_{j} E_{ij} \\ > & \big( e_{i}(t) - e_{j}(t) \big) E_{ij}. > \end{align}$ > And $\begin{align} > t \cdot (E_{i i}- E_{i+1, i+1}) & = t \cdot E_{i i} - t \cdot E_{i+1, i+1} \\ > &= 0 . > \end{align}$ > This gives $\mathfrak{g}_{0}=\mathfrak{t}$ a simult. eigenbasis of the $n-1$ basis vectors $E_{ii}-E_{i+1,i+1}$ and otherwise get $n^{2}-n$ root spaces $\mathfrak{g}_{e_{i}-e_{j}}$, $i \neq j$, corresponding to the basis vectors $E_{ij}$. ----- #### - [[special linear Lie subalgebra]] ---- #### 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 > ```