John Stillwell - Naive Lie Theory.pdf - Index of

3
Generalized rotation groups
PREVIEW
In this chapter we generalize the plane and space rotation groups SO(2)
and SO(3) to the special orthogonal group SO(n) of orientation-preserving
isometries of R n that fix O. To deal uniformly with the concept of “rotation”
in all dimensions we make use of the standard inner product on R n
and consider the linear transformations that preserve it.
Such transformations have determinant +1 or−1 according as they
preserve orientation or not, so SO(n) consists of those with determinant 1.
Those with determinant ±1makeupthefullorthogonal group, O(n).
These ideas generalize further, to the space C n with inner product defined
by
(u 1 ,u 2 ,...,u n ) · (v 1 ,v 2 ,...,v n )=u 1 v 1 + u 2 v 2 + ···+ u n v n . (*)
The group of linear transformations of C n preserving (*) is called the unitary
group U(n), and the subgroup of transformations with determinant 1
is the special unitary group SU(n).
There is one more generalization of the concept of isometry—to the
space H n of ordered n-tuples of quaternions. H n has an inner product defined
like (*) (but with quaternion conjugates), and the group of linear
transformations preserving it is called the symplectic group Sp(n).
In the rest of the chapter we work out some easily accessible properties
of the generalized rotation groups: their maximal tori, centers, and their
path-connectedness. These properties later turn out to be crucial for the
problem of identifying simple Lie groups.
48 J. Stillwell, Naive Lie Theory, DOI: 10.1007/978-0-387-78214-0 3,
c○ Springer Science+Business Media, LLC 2008