Chapter 8: Symplectic Groups
Symplectic Transformations
Orthogonal and unitary transformations preserve symmetric inner products; symplectic transformations preserve an antisymmetric product. Let $\Omega$ be the $2m\times2m$ matrix with block structure \begin{equation} \Omega = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix} \end{equation} where $I_m$ denotes the $m\times m$ identity matrix. Then the real symmetric groups $\Sp(2m,\RR)$ are defined by 1) \begin{equation} \Sp(2m,\RR) = \{ M\in\RR^{2m\times 2m} : M\Omega M^T = \Omega \} \end{equation} Although not obvious at the group level, 2) \begin{equation} M\Omega M^T = \Omega \Longleftrightarrow M^T\Omega M = \Omega \end{equation} so an equivalent definition is \begin{equation} \Sp(2m,\RR) = \{ M\in\RR^{2m\times 2m} : M^T\Omega M = \Omega \} \end{equation}
Other real forms of the symplectic groups can be obtained by first complexifying $\Sp(2m,\RR)$, that is, by considering 3) \begin{equation} \Sp(2m,\RR)\otimes\CC = \{ M\in\CC^{2m\times 2m} : M^T\Omega M = \Omega \} \end{equation} Of particular interest is the compact real form, obtained as the intersection of $\Sp(2m,\RR)\otimes\CC$ with $\SU(2m)$, that is, the groups \begin{equation} \Sp(m) = \{ M\in\CC^{2m\times 2m} : M^T\Omega M = \Omega, M^\dagger M = I \} \end{equation} The dimension of the symplectic groups is given by \begin{equation} |\Sp(m)| = m(2m+1) \end{equation} Real forms with different signatures can be obtained by intersecting with $\SU(p,q)$, with $p+q=2m$, rather than with $\SU(2m)$.
1)
There are many different, overlapping conventions for the names of the
symplectic groups. The group $\Sp(2m,\RR)$ is also written as $\Sp(m,\RR)$.
Since $i\Omega$ is a Hermitian inner product of signature $(m,m)$,
$\Sp(2m,\RR)$ is a real subgroup of $\SU(m,m)$, and is also written as
$\Sp(m,m)$.
2)
This result is easy to see at the Lie algebra level, that is, by treating $M$
as a 1-parameter family of transformations connected to the identity, and
differentiating with respect to the parameter. Denoting the derivative of $M$
at the identity element by $A$, we have
$A\Omega + \Omega A^T = 0$
if $M\in\Sp(2m,\RR)$. Multiplying on both sides by $\Omega$, and using the
fact that $\Omega^2=-I$ results in
$\Omega A+A^T \Omega = 0$
and we have successfully moved the transposed matrix from one side to the other.
3)
Again, there are several conventions for the name of this group, and for the
names of its subgroups. However, we avoid the name $\Sp(2m,\CC)$, which we
reserve for a different group.
