   Next: Picturing the Correspondences Up: Lie Algebras Previous: The Adjoint Representation

#### Unitary Matrices: Now for a different example. is the group of unitary matrices, i.e., complex matrices satifying . An easy computation shows that . is the subgroup for which the determinant is 1 (unimodular matrices). Unlike the situation with and , the dimensions of and (as manifolds) differ by 1.

The Lie algebras of and are denoted and , respectively. Differentiating we conclude that consists of anti-Hermitian matrices: . Note that is anti-Hermitian if and only if is Hermitian.

Fact: if , then (where tr = trace). (Expanding by minors does the trick.) This makes one half of the following fact obvious: the Lie algebra for the Lie group of unimodular matrices consists of all the traceless matrices.

For the special case , things work out very nicely. Since , one can write down the components for easily, and equating them to , one concludes that consists of all matrices of the form:  defining i,j,kas the given matrices in . Exercise: these four elements satisfy the multiplication table of the quaternions, so is isomorphic to the group of quaternions of norm 1. (The somewhat peculiar arrangement of in the displayed element of is dictated by convention.)

Next, an arbitary anti-Hermitian matrix looks like:  This is traceless if and only if . So we have a canonical 1-1 correspondence between and , and so also with : .

It turns out that this correspondence is a Lie algebra isomorphism. and are locally isomorphic, but not isomorphic-- as we will see next. acts on via the adjoint representation. But we have a 1-1 correspondence between and , so we have a representation of in the group of real matrices. Let be an element of and be an element of . Note that . Since the map preserves determinants, it preserves norms when considered as acting on . So the adjoint representation maps into . Fact: it maps onto .

Incidentally, you can see directly that preserves anti-Hermiticity by writing it .

The kernel of the adjoint representation for is its center, which clearly contains -- and in fact, consists of just those two elements. So we have a 2-1 mapping . Our double cover! I'll look at the topology of this in a moment.

Physicists prefer to work with the Pauli spin matrices instead of the quaternions. The Pauli matrices are just the Hermitian counterparts to i, j, and k: , , They form a basis (with 1) for the vector space of Hermitian matrices:   acts on the space of traceless Hermitian matrices in the same way as on : .   Next: Picturing the Correspondences Up: Lie Algebras Previous: The Adjoint Representation