General fact: for any Lie group , there is a homomorphism (also known as a representation) of into the group of non-singular linear transformations on the vector space , with kernel , the center of .
Here's how it goes. For any group , we have the group of inner automorphism and a homomorphism defined by , where . The kernel is . The automorphism is furthermore determined completely by its effects on any set of generators for .
Now take to be a Lie group. Let's consider the effect of on
an ``infinitesimal'' generator
, where .
For , this is rather intuitive. Suppose is a rotation about the axis determined by vector . Then is a rotation about the axis : . If we think of as an infinitesimal rotation, then we see that the action of on given by looks just like the action of on .
Only in three dimensions do things work out so neatly. is abelian, and the adjoint representation for abelian Lie groups is boring-- the trivial homomorphism. And the vector space has dimension 6, so the adjoint representation gives an imbedding of in the group of non-singular matrices.
© 2001 Michael Weiss