John Stillwell - Naive Lie Theory.pdf - Index of

112 5 The tangent space
However, there is no “sl(n,H)” of quaternion matrices of trace zero.
This set of matrices is closed under sums and scalar multiples but, because
of the noncommutative quaternion product, not under the Lie bracket. For
example, we have the following matrices of trace zero in M 2 (H):
( ) ( )
i 0
j 0
X = , Y = .
0 −i 0 −j
But their Lie bracket is
XY −YX =
( ) ( ) ( )
k 0 −k 0 k 0
−
= 2 ,
0 k 0 −k 0 k
which does not have trace zero.
The quaternion Lie algebra that interests us most is sp(n), the tangent
space of Sp(n). As we found in Section 5.3,
sp(n)={X ∈ M n (H) : X + X T = 0},
where X denotes the result of replacing each entry of X by its quaternion
conjugate.
There is no neat relationship between sp(n) and gl(n,H) analogous
to the relationship between su(n) and sl(n,C). This can be seen by considering
dimensions: gl(n,H) has dimension 4n 2 over R, whereas sp(n)
has dimension 2n 2 + n, as we saw in Section 5.5. Therefore, we cannot
decompose gl(n,H) into two subspaces that look like sp(n), because the
dimensions do not add up.
As a result, we need to analyze sp(n) from scratch, and it turns out to
be “simpler” than gl(n,H), in a sense we will explain in Section 6.6.
Exercises
5.7.1 Give three examples of subspaces of gl(n,H) closed under the Lie bracket.
5.7.2 What are the dimensions of your examples?
5.7.3 If your examples do not include one of real dimension 1, give such an example.
5.7.4 Also, if you have not already done so, give an example g of dimension n
that is commutative. That is, [X,Y]=0 for all X,Y ∈ g.