4.17. Lecture 17

4.17.1. The Lie algebra $\mathfrak{sl}_{3,\mathbb{C}}$

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 analogue of the standard basis $H,X,Y$ of $\mathfrak{sl}_{2,\mathbb{C}}$ . We denote by $E_{ij}$ 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.

Proposition 4.17.1.

The set $\mathfrak{h}\subseteq\mathfrak{g}$ , given by

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

is an abelian subalgebra.

Proof.

It is straightforward to see this is a subalgebra as it is closed under scalar multiplication, addition and the Lie bracket. It is abelian as diagonal matrices commute with each other. ∎

We call $\mathfrak{h}$ the (standard) Cartan subalgebra of $\mathfrak{g}$ . We pick as a basis of $\mathfrak{h}$ the elements

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

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, i.e.

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

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 $\mathfrak{h}$ .

4.17.2. Weights

Suppose that $(\rho,V)$ is a finite-dimensional complex-linear representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Suppose that ${\bf v}\in V$ is an eigenvector for all $\rho(H)$ , $H\in\mathfrak{h}$ (called a simultaneous eigenvector). Then, for each $H\in\mathfrak{h}$ , there is an $\alpha(H)\in\mathbb{C}$ such that $\rho(H){\bf v}=\alpha(H)v$ . Since $\rho$ is complex-linear, $\alpha$ is a complex-linear map $\mathfrak{h}\rightarrow\mathbb{C}$ . In other words, $\alpha$ is an element of the dual space $\mathfrak{h}^{*}$ . This motivates the following definition:

Definition 4.17.2.

Suppose that $(\rho,V)$ is a representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Then a weight vector in $V$ is ${\bf v}\in V$ such that there is $\alpha\in\mathfrak{h}^{*}$ (the weight) with:

\rho(H){\bf v}=\alpha(H){\bf v}

for all $H\in\mathfrak{h}$ .

The weight space of $\alpha$ is

V_{\alpha}=\{v\in V\,|\,\rho(H){\bf v}=\alpha(H){\bf v}\text{ for all $H\in% \mathfrak{h}$}\}.

On $\mathfrak{h}$ we have some ‘obvious’ maps $L_{i}\in\mathfrak{h}^{*}$ given by

L_{i}\left(\begin{smallmatrix}a_{1}&&\\ &a_{2}&\\ &&a_{3}\end{smallmatrix}\right)=a_{i}.

These span $\mathfrak{h}^{*}$ , subject to the relation¹¹ 1 More precisely, $\mathfrak{h}^{*}$ is isomorphic to the quotient of the three dimensional vector space with basis $\{L_{1},L_{2},L_{3}\}$ by the subspace spanned by $L_{1}+L_{2}+L_{3}$ .

L_{1}+L_{2}+L_{3}=0.

We compute the weights of some particular representations.

Example 4.17.3.

If $V=\mathbb{C}^{3}$ is the standard representation of $\mathfrak{sl}_{3,\mathbb{C}}$ (for which $\rho(A)=A$ for all $A\in\mathfrak{sl}_{3,\mathbb{C}}$ ), then the standard basis vectors ${\bf e}_{1},{\bf e}_{2},{\bf e}_{3}$ are all weight vectors:

\begin{pmatrix}a_{1}&&\\ &a_{2}&\\ &&a_{3}\end{pmatrix}{\bf e}_{i}=a_{i}{\bf e}_{i}

from which we see that $H{\bf e}_{i}=L_{i}(H){\bf e}_{i}$ for all $H\in\mathfrak{h}$ . Hence ${\bf e}_{i}$ is a weight vector with weight $L_{i}$ .

Example 4.17.4.

If $V=(\mathbb{C}^{3})^{*}$ is the dual of the standard representation then it has a basis ${\bf e}_{1}^{*},{\bf e}_{2}^{*},{\bf e}_{3}^{*}$ defined by

{\bf e}_{i}^{*}({\bf e}_{j})=\delta_{ij}.

One can show that ${\bf e}_{i}^{*}$ is a weight vector of weight $-L_{i}$ , so the weights are $-L_{1},-L_{2},-L_{3}$ . See problem 4.17.3.

Example 4.17.5.

Let $V=\operatorname{Sym}^{2}(\mathbb{C}^{3})$ be the symmetric square of the standard representation. The rules for calculating the weights of $V$ are the same as for $\mathfrak{sl}_{2,\mathbb{C}}$ — so, for the symmetric square, we add all unordered pairs of weights of $\mathbb{C}^{3}$ . For details see section 4.18.3 The weights of $\mathbb{C}^{3}$ are $\{L_{1},L_{2},L_{3}\}$ and so the weights of $\operatorname{Sym}^{2}(\mathbb{C}^{3})$ are

\{2L_{1},2L_{2},2L_{3},L_{1}+L_{2},L_{2}+L_{3},L_{1}+L_{3}\}.

Note that, if we wanted, we could also write $L_{1}+L_{2}=-L_{3}$ etc.

Example 4.17.6.

Let $V=\mathfrak{g}$ with the adjoint representation defined by $\rho(X)Y=[X,Y]$ . As already observed, we have

[H,E_{ij}]=L_{i}-L_{j}

for $H\in\mathfrak{h}$ and $i\neq j$ , while $[H,H^{\prime}]=0$ for $H,H^{\prime}\in\mathfrak{h}$ . Thus the weights of the adjoint representation are $L_{i}-L_{j}$ ( $i\neq j$ ) and $0$ . The weight space for $0$ is $\mathfrak{h}$ , which has dimension two with basis $H_{12}$ and $H_{23}$ ; we say that the weight $0$ has multiplicity two in $V$ . We obtain:

\begin{array}[h]{c|ccc}\text{Weight}&\text{Weight space basis}&&\\ \cline{1-2}\cr\ldelim.{2}{0mm}\ldelim.{3}{0mm}L_{1}-L_{2}&E_{12}&\rdelim\}{3}{% *}\hbox{\multirowsetup\text{Positive roots}}&\rdelim\}{2}{*}\hbox{% \multirowsetup Simple roots}\\ L_{2}-L_{3}&E_{23}\\ L_{1}-L_{3}&E_{13}&\\ \cline{1-2}\cr\ldelim.{3}{0mm}L_{2}-L_{1}&E_{21}&\rdelim\}{3}{*}\hbox{% \multirowsetup\text{Negative roots}}&\\ L_{3}-L_{2}&E_{32}&\\ L_{3}-L_{1}&E_{31}&\\ \cline{1-2}\cr 0&H_{12},H_{13}&&\end{array}

Definition 4.17.7.

A root of $\mathfrak{g}=\mathfrak{sl}_{3,\mathbb{C}}$ is a non-zero weight of the adjoint representation. A root vector is a weight vector of a root, and a root space is the weight space of a root.

In other words, a root $\alpha$ with root vector $0\neq E\in\mathfrak{g}$ is a non-zero element $\alpha\in\mathfrak{h}^{*}$ such that

[H,E]=\alpha(H)E.

We write

\Phi=\{\pm(L_{1}-L_{2}),\pm(L_{2}-L_{3}),\pm(L_{1}-L_{3})\}

for the set of roots of $\mathfrak{sl}_{3,\mathbb{C}}$ . Out of these, we call $\Phi^{+}=\{L_{1}-L_{2},L_{2}-L_{3},L_{1}-L_{3}\}$ the positive roots and $\Phi^{-}=\{L_{2}-L_{1},L_{3}-L_{2},L_{3}-L_{1}\}$ the negative roots. We write $\Delta=\{L_{1}-L_{2},L_{2}-L_{3}\}$ ; these are the simple roots. Note that $L_{1}-L_{3}$ is the sum of the two simple roots. We will sometimes write $\alpha_{ij}$ for the root $L_{i}-L_{j}$ .

Finally, we have the root space or Cartan decomposition

\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha},

where the $\mathfrak{g}_{\alpha}$ are the root spaces, which are all one-dimensional.

4.17.3. Exercises

Problem 46. Verify that

\left[\begin{pmatrix}a_{1}&0&0\\ 0&a_{2}&0\\ 0&0&a_{3}\end{pmatrix},E_{ij}\right]=(a_{i}-a_{j})E_{ij}

and

[E_{12},E_{23}]=E_{13}.

Problem 47. Work through all the theory in Section 4.17.1 for the case of $\mathfrak{sl}_{2,\mathbb{C}}$ . What are the roots and root spaces? What is the relation between the weights (as linear maps on $\mathfrak{h}$ ) and between the weights defined in section 3?

Problem 48. Let $V=(\mathbb{C}^{3})^{*}$ be the dual of the standard representation, with basis ${\bf e}_{1}^{*},{\bf e}_{2}^{*},{\bf e}_{3}^{*}$ dual to the standard basis.

(a)

Show that the ${\bf e}_{i}^{*}$ are weight vectors with weights $-L_{i}$ .
(b)

Find the action of each $E_{ij}$ on ${\bf e}_{3}^{*}$ and deduce that ${\bf e}_{3}^{*}$ is a highest weight vector with weight $-L_{3}$ .