John Stillwell - Naive Lie Theory.pdf - Index of

5.8 Discussion 115
took the trouble to find the centers of various groups in Chapter 3. It turns
out, as we will show in Chapter 7, that g can “see” all the normal subgroups
of G except those that lie in the center, so in finding the centers we have
already found all the normal subgroups.
The pioneers of Lie theory, such as Lie himself, were not troubled by
the subtle difference between simplicity of a Lie group and simplicity of its
Lie algebra. They viewed Lie groups only locally and took members of the
Lie algebra to be members of the Lie group anyway (the “infinitesimal” elements).
For the pioneers, the problem was to find the simple Lie algebras.
Lie himself found almost all of them, as Lie algebras of classical groups.
But finding the remaining simple Lie algebras—the so-called exceptional
Lie algebras—was a monumentally difficult problem. Its solution by Wilhelm
Killing around 1890, with corrections by Élie Cartan in 1894, is now
viewed as one of the greatest achievements in the history of mathematics.
Since the 1920s and 1930s, when Lie groups came to be viewed as
global objects and Lie algebras as their tangent spaces at 1, the question of
what to say about simple Lie groups has generally been ignored or fudged.
Some authors avoid saying anything by defining a simple Lie group to be
one whose Lie algebra is simple, often without pointing out that this conflicts
with the standard definition of simple group. Others (such as Bourbaki
[1972]) define a Lie group to be almost simple if its Lie algebra is
simple, which is another way to avoid saying anything about the genuinely
simple Lie groups.
The first paper to study the global properties of Lie groups was Schreier
[1925]. This paper was overlooked for several years, but it turned out to
be extremely prescient. Schreier accurately identified both the general role
of topology in Lie theory, and the special role of the center of a Lie group.
Thus there is a long-standing precedent for studying Lie group structure as
a topological refinement of Lie algebra structure, and we will take up some
of Schreier’s ideas in Chapters 8 and 9.