In § 8.1 we showed that $\Sp(2)\cong\Spin(5)$, the double cover of $\SO(5)$, and in § 9.2 we showed that $\SU(2,\HH)\cong\Spin(5)$. Meanwhile, in § 8.2 we showed that $\Sp(4,\RR)\cong\Spin(3,2)$, the double cover of $\SO(3,2)$. Is there a relationship between symplectic groups and the quaternions?

We first ask under what conditions complex matrices can be reinterpreted as $m\times m$ quaternionic matrices. It is easier to go the other way: How do we turn a quaternionic matrix into a complex matrix?

Begin with the simplest case: How do we turn a quaternion into $2\times2$ complex matrix? That's easy: Use ($\pm i$ times the) Pauli matrices to represent the quaternionic units. Thus, the matrix representation of the quaternion $a + bi + cj + dk$ is given by \begin{equation} a I + b (i\sigma_z) - c (i\sigma_y) + d (i\sigma_x) = \begin{pmatrix} a+bi & -c+di \\ c+di & a-bi \end{pmatrix} \label{HtoC} \end{equation} This decomposition is just the Cayley-Dickson process in reverse!

What properties do such matrices have? It's not hard to see that a complex matrix $M$ has the form given in (\ref{HtoC}) if and only if \begin{equation} \bar{M} = -\Omega M \Omega \label{CisH} \end{equation} where $M$ and $\Omega$ are now $2\times2$ matrices, that is, $m=1$. In an appropriate basis, however, nothing changes; a $2m\times2m$ complex matrix that satisfies (\ref{CisH}) can be reinterpreted as an $m\times m$ quaternionic matrix.

If $M$ is complex and satisfies both $M^T\Omega M=\Omega$ and $\bar{M}=-\Omega M\Omega$, then \begin{equation} M^\dagger M = \bar{M}^T M = (-\Omega M\Omega)^T M = -\Omega M^T \Omega M = -\Omega \Omega = I \label{spH} \end{equation} and we have shown that \begin{equation} \SU(m,\HH) \cong \Sp(m) \end{equation} The case $m=1$ is just \begin{equation} \Sp(1) \cong \SU(1,\HH) \cong \Spin(3) \end{equation} which is just the isometry group of the quaternions, and, as we have already seen, \begin{equation} \Sp(2) \cong \SU(2,\HH) \cong \Spin(5) \end{equation} when $m=2$.

A similar argument can be used to show that $\SU(p,q,\HH)$ is a symplectic group, often written as $\Sp(p,q,\RR)$. 1) Thus, some symplectic groups can be reinterpreted as quaternionic unitary groups. What about $\Sp(2m,\RR)$?

Consider $\Sp(4,\HH)$, which is the double cover of $\SO(3,2)$. Thus, $\Sp(4,\HH)$ has $3\times2=6$ boosts. But it is easy to show that $\SU(p,q,\HH)$ has $4pq$ boosts. Thus, $\Sp(4,\HH)\not\cong\SU(p,q,\HH)$ for any $p+q=5$. However, $\Sp(4,\HH)$ is referred to as the “split” form of $\Sp(2)$, which suggests that we should use the split form of the quaternions, namely $\HH'$.

So what is $\Sp(m,\HH')$? Again, we start with the case $m=1$, and again we use Pauli matrices. The split quaternion $a + bL + cK + dKL$ can be represented by \begin{equation} a \sigma_t + b \sigma_z - c (i\sigma_y) + d \sigma_x = \begin{pmatrix} a+b & -c+d \\ c+d & a-b \end{pmatrix} \label{HptoC} \end{equation} where we have written $\sigma_t$ rather than $I$ for the $2\times2$ identity matrix. We want to repeat the computation in (\ref{spH}), but we must now carefully distinguish between conjugation in $\HH'$ and conjugation in $\CC$.

Let's start again. According to the Cayley-Dickson process discussed in § 5.1, we can combine a pair of complex numbers $a,b\in\CC$ into a matrix \begin{equation} q = \begin{pmatrix} a & -b \\ \bar{b}\epsilon & \bar{a} \\ \end{pmatrix} \end{equation} where $\epsilon=1$ if $q\in\HH$, and $\epsilon=-1$ if $q\in\HH'$. We can generalize this construction to matrices, leading to the representation \begin{equation} Q = \begin{pmatrix} A & -B \\ \bar{B}\epsilon & \bar{A} \\ \end{pmatrix} \end{equation} of a (possibly split) quaternionic matrix in terms of complex matrices $A$ and $B$. As in the Cayley-Dickson process, the (quaternionic!) conjugate of $Q$ is obtained by replacing $A$ with $\bar{A}$, and $B$ with $-B$, and the (quaternionic!) transpose of $Q$ is obtained by replacing each of $A$ and $B$ with its transpose. We therefore have \begin{equation} Q^\dagger = \begin{pmatrix} A^\dagger & B^T \\ -B^\dagger\epsilon & A^T \\ \end{pmatrix} = -\Omega M^T \Omega \end{equation} which however only reduces to the complex Hermitian conjugate of $Q$ if $\epsilon=1$. Using quaternionic conjugation throughout, the computation in (\ref{spH}) now goes through unchanged whether the components of $M$ are in $\HH$ or $\HH'$, and we have shown that \begin{equation} \SU(m,\HH') \cong \Sp(2m,\RR) \end{equation} Thus, all of the (real forms of the) symplectic groups are indeed quaternionic, but possibly split.

We have just seen that the real symplectic groups are really the quaternionic generalization of the unitary groups. But we can also construct symplectic groups over other division algebras besides the reals.

The definition of symplectic groups given in § 8.1 is normally used verbatim to define symplectic groups over other division algebras. However, we follow Sudbery in using Hermitian conjugation, rather than transpose, in the definition of generalized symplectic groups. That is, we define \begin{equation} \Sp(2m,\KK) = \{ M\in\KK^{2m\times 2m} : M\Omega M^\dagger = \Omega \} \end{equation} for $\KK=\RR,\CC,\HH,\OO$. If $\KK=\CC$, $i\Omega$ is a Hermitian product of signature $(m,m)$. In this case, we can identify these symplectic groups with unitary groups, namely \begin{equation} \Sp(2m,\CC) \cong \SU(m,m) \end{equation} and in particular \begin{align} \Sp(4,\CC) &\cong \SU(2,2) \\ \Sp(6,\CC) &\cong \SU(3,3) \end{align}