1. The group of complex matrices in $GL(2n,\mathbb{C})$ that leaves invariant the anti-symmetric form
\begin{equation}
\sum_{i}^{n}z_i\tilde{w}_i-\tilde{z}_iw_i=\sum_{i}^{n}z_i'\tilde{w}'_i-\tilde{z}'_iw'_i
\end{equation}
of the complex vectors $\vec{z} = (z_1,...,z_n,\tilde{z}_1,...,\tilde{z}_n)$ y $\vec{w} = (w_1,...,w_n,\tilde{w}_1,...,\tilde{w}_n)$ form the symplectic complex group $Sp(2n,\mathbb{C})$ of $2n(2n+1)$ parameters. What is the form of the metric $g$ and what condition must the transformation matrices A satisfy? If the matrices are real, the group is the real symplectic group $Sp(2n, R)$ of $n(2n + 1)$ parameters. If the matrices are unitary, the group is the unitary symplectic group $Sp(2n)$ also with $n(2n + 1)$ parameters.
\begin{itemize}
\item Show that there is a one to one correspondence between the unitary symplectic group $ Sp(2) $ and $ SU(2) $, that is, that they are isomorphic.
\item Consider the group of unitary symplectic transformations in $ 2n $ dimensions $ Sp(2) $. If the infinitesimal generators $ X_k $ are expressed in terms of $ n \times n $ sub-matrices, $X_{ij}$ as
\begin{equation}
X_k=
\begin{pmatrix}
X_{11} & X_{12} \\
X_{21} & X_{22}
\end{pmatrix}
\end{equation}
i) What conditions must the $ X_{ij} $ matrices satisfy?
ii) Show that there are $ r = n(2n+1) $ infinitesimal generators.
\item A generic form of the generators in terms of the $ E_{ij} $ matrices is given by
\begin{equation}
B_{1j}=
\begin{pmatrix}
E_{ij} & 0 \\
0 & -E_{ji}
\end{pmatrix},\,\,\,\,\,\,
C_{ij}=
\begin{pmatrix}
0 & E_{ij}+E_{ji} \\
0 & 0
\end{pmatrix},\,\,\,\,\,\,
D_{ij}=
\begin{pmatrix}
0 & 0 \\
E_{ij}+E_{ji} & 0
\end{pmatrix},
\end{equation}
What are the structure constants?
\item Show that the quadratic Casimir operator is given by
\begin{equation}
C_{2Sp(2n)}=\sum_{ij}B_{ij}B_{ji}+\frac{1}{2}\sum_{ij}(C_{ij}D_{ji}+D_{ij}C_{ji}).
\end{equation}
\end{itemize}