John Stillwell - Naive Lie Theory.pdf - Index of

2
Groups
PREVIEW
This chapter begins by reviewing some basic group theory—subgroups,
quotients, homomorphisms, and isomorphisms—in order to have a basis
for discussing Lie groups in general and simple Lie groups in particular.
We revisit the group S 3 of unit quaternions, this time viewing its relation
to the group SO(3) as a 2-to-1 homomorphism. It follows that S 3 is
not a simple group. On the other hand, SO(3) is simple, asweshowbya
direct geometric proof.
This discovery motivates much of Lie theory. There are infinitely many
simple Lie groups, and most of them are generalizations of rotation groups
in some sense. However, deep ideas are involved in identifying the simple
groups and in showing that we have enumerated them all.
To show why it is not easy to identify all the simple Lie groups we
make a special study of SO(4), the rotation group of R 4 . Like SO(3),
SO(4) can be described with the help of quaternions. But a rotation of
R 4 generally depends on two quaternions, and this gives SO(4) a special
structure, related to the direct product of S 3 with itself. In particular, it
follows that SO(4) is not simple.
J. Stillwell, Naive Lie Theory, DOI: 10.1007/978-0-387-78214-0 2, 23
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