Loop algebra

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.
Definition.
If is a Lie algebra, the tensor product of with , the algebra of (complex) smooth functions over the circle manifold (equivalently, smooth complex-valued periodic functions of a given period),
is an infinite-dimensional Lie algebra with the Lie bracket given by
Here and are elements of and and are elements of .
This isn't precisely what would correspond to the direct product of infinitely many copies of , one for each point in , because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from to ; a smooth parametrized loop in , in other words. This is why it is called the loop algebra.
Loop group.
Similarly, a set of all smooth maps from to a Lie group forms an infinite-dimensional Lie group (Lie group in the sense we can define functional derivatives over it) called the loop group. The Lie algebra of a loop group is the corresponding loop algebra.
Fourier transform.
We can take the Fourier transform on this loop algebra by defining
as
where
is a coordinatization of .
Applications.
If is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Lie algebra.