John Stillwell - Naive Lie Theory.pdf - Index of

3.4 The symplectic groups 57
3.4 The symplectic groups
On the space H n of ordered n-tuples of quaternions there is a natural inner
product,
(p 1 , p 2 ,...,p n ) · (q 1 ,q 2 ,...,q n )=p 1 q 1 + p 2 q 2 + ···+ p n q n . (**)
This of course is formally the same as the inner product (*) on C n ,except
that the p i and q j now denote arbitrary quaternions. The space H n
is not a vector space over H, because the quaternions do not act correctly
as “scalars”: multiplying a vector on the left by a quaternion is in general
different from multiplying it on the right, because of the noncommutative
nature of the quaternion product.
Nevertheless, quaternion matrices make sense (thanks to the associativity
of the quaternion product, we still get an associative matrix product),
and we can use them to define linear transformations of H n . Then, by specializing
to the transformations that preserve the inner product (**), we get
an analogue of the orthogonal group for H n called the symplectic group
Sp(n). As with the unitary groups, preserving the inner product implies
preserving length in the corresponding real space, in this case in the space
R 4n corresponding to H n .
For example, Sp(1) consists of the 1 × 1 quaternion matrices, multiplication
by which preserves length in H = R 4 . In other words, the members
of Sp(1) are simply the unit quaternions. Because we defined quaternions
in Section 1.3 as the 2 × 2 complex matrices
( )
a + id −b − ic
,
b − ic a − id
it follows that
{( )
}
a + id −b − ic
Sp(1)=
: a 2 + b 2 + c 2 + d 2 = 1 = SU(2).
b − ic a − id
Thus we have already met the first symplectic group.
The quaternion matrices A in Sp(n), like the complex matrices in
SU(n), are characterized by the condition AA T = 1, where the bar now
denotes the quaternion conjugate. The proof is the same as for SU(n).
Because of this formal similarity, there is a proof that Sp(n) is pathconnected,
similar to that for SU(n) given in the previous section.