John Stillwell - Naive Lie Theory.pdf - Index of

202 9 Simply connected Lie groups
Lie group G. This result strongly constrains the topology of Lie groups,
because the fundamental group of an arbitrary smooth manifold can be any
finitely presented group. A “random” smooth manifold has a nonabelian
fundamental group.
Like the quotient construction (see Section 3.9), the universal covering
can produce a nonmatrix group ˜G from a matrix group G. A famous
example, essentially due to Cartan [1936], is the universal covering group
˜ SL(2,C) of the matrix group SL(2,C). Thus topology provides another
path to the world of Lie groups beyond the matrix groups.
Topology makes up the information lost when we pass from Lie groups
to Lie algebras, and in fact topology makes it possible to bypass Lie algebras
almost entirely. A notable book that conducts Lie theory at the group
level is Adams [1969], by the topologist J. Frank Adams. It should be said,
however, that Adams’s approach uses topology that is more sophisticated
than the topology used in this chapter.
Finite simple groups
The classification of simple Lie groups by Killing and Cartan is a remarkable
fact in itself, but even more remarkable is that it paves the way for the
classification of finite simple groups—a much harder problem, but one that
is related to the classification of continuous groups. Surprisingly, there are
finite analogues of continuous groups in which the role of R or C is played
by finite fields. 11
As mentioned in Section 2.8, finite simple groups were discovered by
Galois around 1830 as a key concept for understanding unsolvability in the
theory of equations. Galois explained solution of equations by radicals as a
process of “symmetry breaking” that begins with the group of all symmetries
of the roots and factors it into smaller groups by taking square roots,
cube roots, and so on. The process first fails with the general quintic equation,
where the symmetry group is S 5 , the group of all 120 permutations of
five things. The group S 5 may be factored down to the group A 5 of the 60
even permutations of five things by taking a suitable square root, but it is
not possible to proceed further because A 5 is a simple group.
More generally, A n is simple for n > 5, so Galois had in fact discovered
an infinite family of finite simple groups. Apart from the infinite family of
11 This brings to mind a quote attributed to Stan Ulam: The infinite we can do right away,
the finite will take a little longer.