The goal is to give the Dynkin diagram of $SU(n)$. One can show that the complexification of the Lie algebra $\mathfrak{g}$ of $G$ is given by $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ (thus the traceless matrices). Moreover one can show that if $\mathfrak{t}$ is the set of all diagonal matrices, that then $\mathfrak{t}$ is a maximal torus of $\mathfrak{g}$. The first step we have to make is the determine what the root system $R=R(\mathfrak{g}_{\mathbb{C}},\mathfrak{t})$ is. Therefore define $\epsilon_k:\mathfrak{t}\rightarrow i\mathbb{R}$ by $\epsilon_k(X)=X_{kk}$ (thus the map takes the $k^{th}$ diagonal element of $X$. Furthermore let $E_{ij}$ $(i\neq j)$ be the matrix with $1$ on the $(i,j)$-place and zero else.
The first claim we make is that $\mathbb{C}E_{ij}$ is a root space. Thus we have to find a linear functiona $\alpha$ such that $[H,\mu E_{ij}]=\alpha(H)\mu E_{ij}$ for $H\in\mathfrak{t}$ and $\mu\in\mathbb{C}$. I thought that this functional has to be the following $\alpha(H)=(\lambda_i-\lambda_j)$ with $\lambda_i,\lambda_j$ the values on the diagonal (and the $i$ and $j$ comming from $E_{ij}$). Thus my idea was that $R=\{\epsilon_i-\epsilon_j:i\neq j\}$. Is this a good idea? If so, then can someone explain how to write this down in a good way, so not where lies the mistake?
The following claim is that $E=i\mathfrak{t}^*$ together $R$ is a root system. This should be doable since the properties of a root system are not quiet difficult.
The next step is to determine a fundamental system $S$ of $R$ and to show what the reflections $s_{\alpha}$ $(\alpha\in S)$ are. Furthermore i hve to know what the Cartan integers of $S$ are to draw the good Dynkin diagram.
For the last steps I need some explainations. Is there someone who can help me with this topic?
Thank you very much.
