In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.
is the space of Laurent polynomials, then
is a Lie algebra, the tensor product of
with C∞(S1), the algebra of (complex) smooth functions over the circle manifold S1 (equivalently, smooth complex-valued periodic functions of a given period),
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 parametrized loop in
This is why it is called the loop algebra.
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
There is a natural derivation on the loop algebra, conventionally denoted
It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.
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.
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.
[1] The central extension is given by adjoining a central element
and modifying the bracket on the loop algebra to
The central extension is, as a vector space,
(in its usual definition, as more generally,
Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra.
Then the extra term added to the bracket is
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[2]
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.