\include{preamble}
\begin{document}
\thispagestyle{empty}
\paragraph{\bf Exercise 3:} Some things about NIBS forms.
\begin{enumerate}
\item Prove that the Killing form is an invariant symmetric bilinear form on any simple finite dimensional complex Lie algebra.
\item Show that the trace form on the standard representation of $\fsl_n$ is non-degenerate.
\item Pick two of the classical types ($A_r, B_r, C_r, D_r$) and calculate how the trace form on the standard representation of each type differs from the Killing form (as a function of $r$). (You'll need a good basis for each to do this.)
\item Let $B=\{b_1, \dots, b_\ell\}$ be a basis for a finite-dimensional reductive complex Lie algebra $\fg$ with a NIBS form $\<,\>$, and define the dual basis
$$B^*=\{b_1^*, \dots, b_\ell^*\} \quad \text{ by } \quad \ = \delta_{i,j}.$$
The \emph{Casimir} element of $\fg$ is
$$\kappa = \sum_{i = 1}^\ell b_ib_i^* \in U \fg.$$
Prove the following.
\begin{enumerate}
\item $\kappa$ does not depend on the choice of basis.
\item $\kappa \in Z(U\fg)$, where $Z(U\fg)$ is the center of $U\fg$ (it suffices to show that $\kappa$ commutes with every element of $\fg$).
\end{enumerate}
[Notice that (i) $B^*$ is also a basis for $\fg$, and (ii) for any basis $B = \{b_i\}_i$ and $x \in \fg$, you have $x = \sum_{i} \ b_i.$]
\end{enumerate}
\end{document}