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Unitary Matrices Up: Lie Algebras

Elements of act on , or equivalently on orthonormal bases of (frames).2 But can also be regarded as a set of transformations on the vector space , as we will see in a moment. Intuitively, the triple (, , ) takes the place of the standard frame for .

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 .

Or in terms of derivatives, if is our prototypical smooth curve through 1, then the derivative of at is . So the vector space is closed under the map . (Remember that both and are sets of matrices.) is clearly contained in the kernel, and in fact: is the kernel.

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.

Next: Unitary Matrices Up: Lie Algebras