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 smooth functions over the circle manifold ,
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 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.