John Stillwell - Naive Lie Theory.pdf - Index of

118 6 Structure of Lie algebras
isasmoothpathinT 1 (H). It likewise follows that
D ′ (0)=XY −YX ∈ T 1 (H),
and hence T 1 (H) is an ideal, as claimed.
Remark. In Section 7.5 we will sharpen this theorem by showing that
T 1 (H) ≠ {0} provided H is not discrete, that is, provided there are points
in H not equal to 1 but arbitrarily close to it. Therefore, if g has no ideals
other than itself and {0}, then the only nontrivial normal subgroups of G
are discrete. We saw in Section 3.8 that any discrete normal subgroup of
a path-connected group G is contained in Z(G), the center of G. For the
generalized rotation groups G (which we found to be path-connected in
Chapter 3, and which are the main candidates for simplicity), we already
found Z(G) in Section 3.7. In each case Z(G) is finite, and hence discrete.
This remark shows that the Lie algebra g = T 1 (G) can “see” normal
subgroups of G that are not too small. T 1 (G) retains an image of a normal
subgroup H as an ideal T 1 (H), which is “visible” (T 1 (H) ≠ {0}) provided
H is not discrete. Thus, if we leave aside the issue of discrete normal
subgroups for the moment, the problem of finding simple matrix Lie groups
essentially reduces to finding the Lie algebras with no nontrivial ideals.
In analogy with the definition of simple group (Section 2.2), we define
a simple Lie algebra to be one with no ideals other than itself and {0}.
By the remarks above, we can make a big step toward finding simple Lie
groups by finding the simple Lie algebras among those for the classical
groups. We do this in the sections below, before returning to Lie groups to
resolve the remaining difficulties with discrete subgroups and centers.
Simplicity of so(3)
We know from Section 2.3 that SO(3) is a simple group, so we do not
really need to investigate whether so(3) is a simple Lie algebra. However,
it is easy to prove the simplicity of so(3) directly, and the proof is a model
for proofs we give for more complicated Lie algebras later in this chapter.
First, notice that the tangent space so(3) of SO(3) at 1 is the same as
the tangent space su(2) of SU(2) at 1. This is because elements of SO(3)
can be viewed as antipodal pairs ±q of quaternions q in SU(2). Tangents
to SU(2) are determined by the q near 1, in which case −q is not near 1,
so the tangents to SO(3) are the same as the tangents to SU(2).
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