John Stillwell - Naive Lie Theory.pdf - Index of

72 3 Generalized rotation groups
Maximal tori were also introduced by Weyl, in his paper Weyl [1925].
In this book we use them only to find the centers of the orthogonal, unitary,
and symplectic groups, since the centers turn out to be crucial in the investigation
of simplicity. However, maximal tori themselves are important for
many investigations in the structure of Lie groups.
The existence of a nontrivial center in SO(2m), SU(n), andSp(n)
shows that these groups are not simple, since the center is obviously a
normal subgroup. Nevertheless, these groups are almost simple, because
the center is in each case their largest normal subgroup. We have shown in
Section 3.8 that the center is the largest normal subgroup that is discrete,
in the sense that there is a minimum, nonzero, distance between any two
of its elements. It therefore remains to show that there are no nondiscrete
normal subgroups, which we do in Section 7.5.
It turns out that the quotient groups of SO(2m), SU(n), andSp(n) by
their centers are simple and, from the Lie theory viewpoint, taking these
quotients makes very little difference. The center is essentially “invisible,”
because its tangent space is zero. We explain “invisibility” in Chapter 5,
after looking at the tangent spaces of some particular groups in Chapter 4.
It should be mentioned, however, that the quotient of a matrix group
by a normal subgroup is not necessarily a matrix group. Thus in taking
quotients we may leave the world of matrix groups. The first example was
discovered by Birkhoff [1936]. It is the quotient (called the Heisenberg
group) of the group of upper triangular matrices of the form
⎛ ⎞
1 x y
⎝0 1 z⎠, where x,y,z ∈ R,
0 0 1
by the subgroup of matrices of the form
⎛
1 0
⎞
n
⎝0 1 0⎠, where n ∈ Z.
0 0 1
The Heisenberg group is a Lie group, but not isomorphic to a matrix group.
One of the reasons for looking at tangent spaces is that we do not have
to leave the world of matrices. A theorem of Ado from 1936 shows that
the tangent space of any Lie group G—the Lie algebra g—can be faithfully
represented by a space of matrices. And if G is almost simple then g is truly