Loop algebra

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

Definition
For a Lie algebra $$\mathfrak{g}$$ over a field $$K$$, if $$K[t,t^{-1}]$$ is the space of Laurent polynomials, then $$L\mathfrak{g} := \mathfrak{g}\otimes K[t,t^{-1}],$$ with the inherited bracket $$[X\otimes t^m, Y\otimes t^n] = [X,Y]\otimes t^{m+n}.$$

Geometric definition
If $$\mathfrak{g}$$ is a Lie algebra, the tensor product of $$\mathfrak{g}$$ with $C^{∞}(S^{1})$, the algebra of (complex) smooth functions over the circle manifold $S^{1}$ (equivalently, smooth complex-valued periodic functions of a given period),

$$\mathfrak{g}\otimes C^\infty(S^1),$$

is an infinite-dimensional Lie algebra with the Lie bracket given by

$$[g_1\otimes f_1,g_2 \otimes f_2]=[g_1,g_2]\otimes f_1 f_2.$$

Here $g_{1}$ and $g_{2}$ are elements of $$\mathfrak{g}$$ and $f_{1}$ and $f_{2}$ are elements of $C^{∞}(S^{1})$.

This isn't precisely what would correspond to the direct product of infinitely many copies of $$\mathfrak{g}$$, one for each point in $S^{1}$, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from $S^{1}$ to $$\mathfrak{g}$$; a smooth parametrized loop in $$\mathfrak{g}$$, in other words. This is why it is called the loop algebra.

Gradation
Defining $$\mathfrak{g}_i$$ to be the linear subspace $$\mathfrak{g}_i = \mathfrak{g}\otimes t^i < L\mathfrak{g},$$ the bracket restricts to a product $$[\cdot\,, \, \cdot]: \mathfrak{g}_i \times \mathfrak{g}_j \rightarrow \mathfrak{g}_{i+j},$$ hence giving the loop algebra a $$\mathbb{Z}$$-graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra $$\mathfrak{g}_0 \cong \mathfrak{g}$$.

Derivation
There is a natural derivation on the loop algebra, conventionally denoted $$d$$ acting as $$d: L\mathfrak{g} \rightarrow L\mathfrak{g}$$ $$d(X\otimes t^n) = nX\otimes t^n$$ and so can be thought of formally as $$d = t\frac{d}{dt}$$.

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 $S^{1}$ 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.

Affine Lie algebras as central extension of loop algebras
If $$\mathfrak{g}$$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra $$L\mathfrak g$$ gives rise to an affine Lie algebra. Furthermore this central extension is unique.

The central extension is given by adjoining a central element $$\hat k$$, that is, for all $$X\otimes t^n \in L\mathfrak{g}$$, $$[\hat k, X\otimes t^n] = 0,$$ and modifying the bracket on the loop algebra to $$[X\otimes t^m, Y\otimes t^n] = [X,Y] \otimes t^{m + n} + mB(X,Y) \delta_{m+n,0} \hat k,$$ where $$B(\cdot, \cdot)$$ is the Killing form.

The central extension is, as a vector space, $$L\mathfrak{g} \oplus \mathbb{C}\hat k$$ (in its usual definition, as more generally, $$\mathbb{C}$$ 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 on the loop algebra. This is the map $$\varphi: L\mathfrak g \times L\mathfrak g \rightarrow \mathbb{C}$$ satisfying $$\varphi(X\otimes t^m, Y\otimes t^n) = mB(X,Y)\delta_{m+n,0}.$$ Then the extra term added to the bracket is $$\varphi(X\otimes t^m, Y\otimes t^n)\hat k.$$

Affine Lie algebra
In physics, the central extension $$L\mathfrak g \oplus \mathbb C \hat k$$ is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space $$\hat \mathfrak{g} = L\mathfrak{g} \oplus \mathbb C \hat k \oplus \mathbb C d$$ where $$d$$ 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.