Loop algebra

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

Definition

For a Lie algebra over a field, if is the space of Laurent polynomials, then
with the inherited bracket

Geometric 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.

Gradation

Defining to be the linear subspace the bracket restricts to a product
hence giving the loop algebra a -graded Lie algebra structure.
In particular, the bracket restricts to the 'zero-mode' subalgebra.

Derivation

There is a natural derivation on the loop algebra, conventionally denoted acting as
and so can be thought of formally as.
It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.

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.

Affine Lie algebras as central extension of loop algebras

If is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Lie algebra. Furthermore this central extension is unique.
The central extension is given by adjoining a central element, that is, for all,
and modifying the bracket on the loop algebra to
where is the Killing form.
The central extension is, as a vector space, .

Cocycle

Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra. This is the map
satisfying
Then the extra term added to the bracket is

Affine Lie algebra

In physics, the central extension is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space
where is the derivation defined above.
On this space, the Killing form can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra.