Loop algebra

Definition

If g is a Lie algebra, the tensor product of g with C^∞(S¹), 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 g and f₁ and f₂ are elements of C^∞(S¹).

This isn't precisely what would correspond to the direct product of infinitely many copies of g, 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 g; a smooth parametrized loop in g, in other words. This is why it is called the loop algebra.

Loop group

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.

Fourier transform

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

as

where

is a coordinatization of S¹.

Applications

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