In the book, a guide to quantum groups. I find a mistake,
the $H_{\alpha}$ should be defined $H_{\alpha}=E_{11}-E_{22}$.
The Casimir element $t_{0}=\sum_{i}\frac{n}{n-1} E_{ii}\otimes E_{ii}-\sum_{i\neq j}\frac{1}{n}E_{ii}\otimes E_{jj}$. You can see the following paper for more detail. http://arxiv.org/pdf/math/9901079v3.pdf .