---- > [!definition] Definition. ([[special linear Lie subalgebra]]) > Let $V$ be an $n$-dimensional [[vector space]] over [[field|field]] $k$. Let $\mathfrak{gl}(V)$ denote the [[general linear Lie algebra]] on $V$. The [[Lie subalgebra]] $\mathfrak{sl}(V):=\{ x \in \mathfrak{gl}(V) : \text{tr }x = 0 \}$ is called the **special linear Lie subalgebra** on $V$. As usual, once a [[basis]] of $V$ is chosen we identify $\mathfrak{sl}(V)$ with a matrix space $\mathfrak{sl}_{n}$. > > The standard [[basis]] for $\mathfrak{sl}(V)$ is $\begin{align} E_{i,j} &\text{ for } i \neq j \\ E_{i,i} - E_{i+1, i+1} &\text{ otherwise } (i < n). \end{align}$ There are $n(n-1)$ matrices of the first type, and $n-1$ of the second. So the space $\mathfrak{sl}_{n}$ is $n^{2}-1$ [[dimension|dimensional]]. ^definition > [!justification] Justification that $\mathfrak{sl}(V)$ is a [[Lie subalgebra]] of $\mathfrak{gl}(V)$. First we show that $\mathfrak{sl}(V)$ is a [[linear subspace]] of $V$; then we show that the bracket $[-,-]_{\mathfrak{g}}$ restricts on it. Clearly the zero map belongs to $\mathfrak{sl}(V)$. Linearity of [[trace of a linear operator|trace]] means that $\mathfrak{sl}(V)$ is stable under addition and scalar multiplication. Finally, let $x,y \in \mathfrak{sl}(V)$. We have $\text{tr }[x, y] = \text{tr}(xy - yx)=\text{tr}(xy)-\text{tr}(yx)=0,$ where in the last step we used that trace is commutative. ^justification > [!basicexample] The most important Lie algebra of all: $\mathfrak{sl}_{2}$. > The most important special linear Lie algebra is$\mathfrak{sl}_{2}=\{ \begin{bmatrix} a& b \\ c & -a \end{bmatrix} : a,b,c \in k\}$ which has as [[basis]] $e=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, f =\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, h = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$ Note $[e,f]=h, [h, e]=2e, [h,f]=-2f.$ (Note that this implies the standard basis of $\mathfrak{sl}_{2}(\mathbb{C})$ is an [[eigenbasis]] for $\text{ad}_{h}=[h,-]$. [[root space decomposition of sln(C)|This holds for all]] $n$.) ^basic-example ---- #### ---- #### 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 > ```