7.1 The Lie algebra sl_3,C

We study the Lie algebra

\mathfrak{g}=\mathfrak{sl}_{3,\mathbb{C}}=\{X\in\mathfrak{gl}_{3,\mathbb{C}}\;%:\;\operatorname{tr}(X)=0\}

of traceless $3\times 3$ matrices. It has dimension $8$ . We first need to find the analog of the standard basis $H, X, Y$ of $\mathfrak{sl}_{2,\mathbb{C}}$

First, notation: we write $E_{ij}$ for the matrix with a ’1’ in row $i$ and column $j$ , and ’0’ elsewhere. Then $E_{ij}\in\mathfrak{sl}_{3,\mathbb{C}}$ if and only if $i\neq j$ .

The analogue of $H$ will be the entire subalgebra of diagonal matrices.

Definition 7.1.

The (standard) Cartan subalgebra of $\mathfrak{g}$ is $\mathfrak{h}$ , given by

\mathfrak{h}=\left\{\begin{pmatrix}a_{1}&&\\&a_{2}&\\&&a_{3}\end{pmatrix}\;:\;a_{1}+a_{2}+a_{3}=0,\right\}.

Note that $\mathfrak{h}$ is an abelian subalgebra, because diagonal matrices commute with each other.

We pick as a basis of $\mathfrak{h}$ the elements

	$\displaystyle H_{12}=E_{11}-E_{22}$	$\displaystyle=\begin{pmatrix}\hidden@noalign{}\hfil\textstyle 1&&\\\hidden@noalign{}\hfil\textstyle&-1&\\\hidden@noalign{}\hfil\textstyle&&0\end{pmatrix}$
and
	$\displaystyle H_{23}=E_{22}-E_{33}$	$\displaystyle=\begin{pmatrix}\hidden@noalign{}\hfil\textstyle 0&&\\\hidden@noalign{}\hfil\textstyle&1&\\\hidden@noalign{}\hfil\textstyle&&-1\end{pmatrix},$

and also define $H_{13}=E_{11}-E_{33}=H_{12}+H_{23}$ .

Next we consider the adjoint action of $\mathfrak{h}$ on $\mathfrak{g}$ , seeking eigenvectors and eigenvalues. The key calculation is:

\left[\begin{pmatrix}a_{1}&&\\&a_{2}&\\&&a_{3}\end{pmatrix},E_{ij}\right]=(a_{i}-a_{j})E_{ij}.

Exercise 7.2.

Check this!

Thus $\{E_{ij}:i\neq j\}\cup\{H_{12},H_{23}\}$ is a basis of simultaneous eigenvectors in $\mathfrak{sl}_{3,\mathbb{C}}$ for the adjoint action of $\mathfrak{h}$ .

7.1 The Lie algebra sl3,C

Definition 7.1.

Exercise 7.2.

7.1 The Lie algebra sl_3,C