More Mathematics . . .

Cyclic Subgroups of the Loop Group of SU(2)

The set of 2 x 2 complex matrices

together with the operation of matrix multiplication forms the three-dimensional, compact, simply connected Lie group SU(2). The algebraic structure of SU(2) also gives rise in a natural way to a metric which makes the group isomorphic as a geometric space to the round 3-sphere. The space of smooth maps from the circle into SU(2) together with the operation of point-wise multiplication also forms a group, the loop group LSU(2). Together with a Frechet topology it can be viewed as an infinite dimensional Lie group. The simplest examples of loops in SU(2) are those conjugate to the diagonal subgroup of SU(2) which determine the geodesics emanating from the identity. Although these loops are easy to write down, their nth-power is merely a loop covering the same subgroup n-times. However, the following loop, which I called for no particular reason, generates a geometrically interesting cyclic subgroup of LSU(2). We write z for the coordinate on the circle .

In the movie below, the first few terms of the cyclic subgroup genererated by are plotted using stereographic coordinates for SU(2). As is the identity in SU(2) when z = 1, all of its powers also start at the identity. The second film shows the first few terms of the cyclic subgroup generated by a translate of by a generic constant loop.