Infinite-Dimensional Lie Groups: Their Geometry, Orbits, and Dynamical Systems

For any finite-dimensional Lie group G one can define the corresponding loop group LG as the set of smooth maps from a circle to the group G endowed with pointwise multiplication.

We are interested in the coadjoint representation of the group LG or, rather, of a central extension LG of the group LG. While the loop group itself turns out to be “too simple” to give rise to a rich theory, its central extension, called the affine (Kac-Moody) group corresponding to G, possesses a beautiful geometry and is related to many other fields in mathematics and mathematical physics. It turns out that the coadjoint orbits of the affine groups have finite codimension and are closely related to their finite-dimensional counterparts. This makes such groups attractive for representation theory: a complete classification of their coadjoint orbits indicates existence of rich representation theory for them, according to Kirillov’s orbit method.

Here we describe the geometric features of loop groups and their extensions. We also comment on more general current groups G^M of smooth maps from a manifold M to a finite-dimensional Lie group G and consider certain classes of such groups in subsequent sections. Throughout this section, G denotes a finite-dimensional connected and simply connected Lie group, and g stands for its Lie algebra.