John Stillwell - Naive Lie Theory.pdf - Index of

6
Structure of Lie algebras
PREVIEW
In this chapter we return to our original motive for studying Lie algebras:
to understand the structure of Lie groups. We saw in Chapter 2 how normal
subgroups help to reveal the structure of the groups SO(3) and SO(4). To
go further, we need to know exactly how the normal subgroups of a Lie
group G are reflected in the structure of its Lie algebra g.
The focus of attention shifts from groups to algebras with the following
discovery. The tangent map from a Lie group G to its Lie algebra g sends
normal subgroups of G to substructures of g called ideals. Thus the ideals
of g “detect” normal subgroups of G in the sense that a nontrivial ideal of
g implies a nontrivial normal subgroup of G.
Lie algebras with no nontrivial ideals, like groups with no nontrivial
normal subgroups, are called simple. It is not quite true that simplicity of
g implies simplicity of G, but it turns out to be easier to recognize simple
Lie algebras, so we consider that problem first.
We prove simplicity for the “generalized rotation” Lie algebras so(n)
for n > 4, su(n), sp(n), and also for the Lie algebra of the special linear
group of C n . The proofs occupy quite a few pages, but they are all variations
on the same elementary argument. It may help to skip the details
(which are only matrix computations) at first reading.
116 J. Stillwell, Naive Lie Theory, DOI: 10.1007/978-0-387-78214-0 6,
c○ Springer Science+Business Media, LLC 2008