The weight lattice

0.00 Suppose we have a compact group $G$ . We know that $G$ contains a maximal torus $T$ . Let $\mathfrak{t}$ be the Lie algebra of $T$ .

Example:

0.55 If $G=SU(3)$ then $\mathfrak{t}=\left\{\begin{pmatrix}i\theta_{1}&0&0\\ 0&i\theta_{2}&0\\ 0&0&-i(\theta_{1}+\theta_{2})\end{pmatrix}\ :\ \theta_{1},\theta_{2}\in\mathbf% {R}\right\}.$

1.15 Let $\mathfrak{h}=\mathfrak{t}\otimes\mathbf{C}\subset\mathfrak{g}\otimes\mathbf{C}$ be the complexification of $\mathfrak{t}$ . In our example, this amounts to allowing $\theta_{1},\theta_{2}$ to be complex. We're really interested in the matrices $H_{\theta}=\begin{pmatrix}\theta_{1}&0&0\\ 0&\theta_{2}&0\\ 0&0&-(\theta_{1}+\theta_{2})\end{pmatrix}$ , and these live inside $\mathfrak{h}_{\mathbf{R}}=i\mathfrak{t}\subset\mathfrak{h}.$

2.30 Given a complex representation $R\colon G\to GL(V)$ , we know that $V=\bigoplus_{\lambda}W_{\lambda}$ where $W_{\lambda}=\{v\in V\ :\ R(\exp i\Theta)=e^{i\lambda(\Theta)}v\ \forall\Theta% \in\mathfrak{h}_{\mathbf{R}}\}.$

3.40 $\lambda$ is a map $\mathfrak{h}_{\mathbf{R}}\to\mathbf{R}$ of the form $\lambda(\Theta)=\lambda_{1}\theta_{1}+\cdots+\lambda_{n}\theta_{n}.$ In other words, $\lambda$ lives in $\mathfrak{h}_{\mathbf{R}}^{*}$ , the dual space of $\mathfrak{h}_{\mathbf{R}}$ . So our weight diagrams are pictures inside $\mathfrak{h}_{\mathbf{R}}$ .

Example:

4.30 For $SU(2)$ , our weight diagram consisted of a line with some dots on. The line was really a picture of $\mathfrak{h}_{\mathbf{R}}=\left\{\begin{pmatrix}\theta&0\\ 0&-\theta\end{pmatrix}\ :\ \theta\in\mathbf{R}\right\}$ . But what were the dots?

5.00 We know that if $\exp(i\Theta)=I$ then $R(\exp(i\Theta))=I$ , so $e^{i\lambda(\Theta)}=1$ .

Definition:

Define $\mathfrak{h}_{\mathbf{Z}}\subset\mathfrak{h}_{\mathbf{R}}$ to be the lattice $\{\Theta\in\mathfrak{h}_{\mathbf{R}}\ :\ \exp(i\Theta)=I\}.$ Then our weights live in the weight lattice $\mathfrak{h}_{\mathbf{Z}}^{*}\subset\mathfrak{h}_{\mathbf{R}}^{*}$ , which is defined to be $\mathfrak{h}_{\mathbf{Z}}^{*}=\{\lambda\in\mathfrak{h}^{*}_{\mathbf{R}}\ :\ % \lambda(\Theta)\in 2\pi\mathbf{Z}\mbox{ whenever }\Theta\in\mathfrak{h}_{% \mathbf{Z}}\}.$

7.25 For example, the triangular dot paper where we drew our weight diagrams of $SU(3)$ representations was a depiction of the weight lattice of $SU(3)$ .

Example:

7.35 For $SU(2)$ , $\mathfrak{h}_{\mathbf{Z}}=\left\{\theta\ :\ \exp\left(i\begin{pmatrix}\theta&0% \\ 0&-\theta\end{pmatrix}\right)=I\right\}=\{2\pi n\ :\ n\in\mathbf{Z}\},$ and $\mathfrak{h}^{*}_{\mathbf{Z}}=\{\lambda\ :\ \lambda 2\pi n\in 2\pi\mathbf{Z}% \quad\forall n\in\mathbf{Z}\}=\mathbf{Z}\subset\mathbf{R}.$ So the weight lattice is really just the integers inside $\mathbf{R}$ .

Example:

8.40 For $SU(3)$ , the weight lattice was the triangular lattice we used to draw pictures on, where the $\lambda_{1}$ and $\lambda_{2}$ "axes" are separated by 120 degrees. In the next couple of videos, we will see that (for a class of Lie algebras called semisimple Lie algebras) the weight lattice comes equipped with a natural geometric structure called the Killing form, and it's with respect to this geometry that the angle is 120 degrees.

Pre-class exercise

Exercise:

Explicitly, what is the set of matrices $\mathfrak{h}_{\mathbf{Z}}$ for our usual diagonal choice of maximal torus in $SU(3)$ ?