Loop Algebra

In mathematics, loop algebras are certain types of Lie algebra, of particular interest in theoretical physics.

If is a Lie algebra, the tensor product of with ,

,

the algebra of (complex) smooth functions over the circle manifold S1 is an infinite-dimensional Lie algebra with the Lie bracket given by

.

Here g₁ and g₂ are elements of and f₁ and f₂ are elements of .

This isn't precisely what would correspond to the direct product of infinitely many copies of, one for each point in S1, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from S1 to ; a smooth parameterized loop in, in other words. This is why it is called the loop algebra.

We can take the Fourier transform on this loop algebra by defining

as

where

0 ≤ σ <2π

is a coordinatization of S1.

If is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Kac-Moody algebra.

Similarly, a set of all smooth maps from S1 to a Lie group G 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.