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

Definition

is the space of Laurent polynomials

Geometric definition

is a Lie algebra, the tensor product 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 and and are elements of .

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

Gradation

to be the linear subspace-graded Lie algebra structure.

.

Derivation

.

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 (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.

Affine Lie algebras as central extension of loop algebras

is a semisimple Lie algebra, then a nontrivial central extension gives rise to an affine Lie algebra. Furthermore, this central extension is unique.

is the Killing form.

can be taken to be an arbitrary field).

Cocycle

Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle

Affine Lie algebra

is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space 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.

References