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 Template:Math, the algebra of (complex) smooth functions over the circle manifold Template:Math (equivalently, smooth complex-valued periodic functions of a given period),

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

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 Template:Math and Template:Math are elements of $\mathfrak{𝔤}$ and Template:Math and Template:Math are elements of Template:Math.

This isn't precisely what would correspond to the direct product of infinitely many copies of $\mathfrak{𝔤}$, one for each point in Template:Math, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from Template:Math 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$$[\cdot ,\cdot ]:{\mathfrak{𝔤}}_i\times {\mathfrak{𝔤}}_j\to {\mathfrak{𝔤}}_{i+j},$$ hence giving the loop algebra a $\mathbb{\mathbb{Z}}$-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 $$d:L\mathfrak{𝔤}\to L\mathfrak{𝔤}$$ $$d(X\otimes t^n)=nX\otimes t^n$$ 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 Template:Math to a Lie group Template:Math 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$$\phi :L\mathfrak{𝔤}\times L\mathfrak{𝔤}\to \mathbb{ℂ}$$ satisfying $$\phi (X\otimes t^m,Y\otimes t^n)=mB(X,Y){\delta }_{m+n,0}.$$ Then the extra term added to the bracket is $\phi (X\otimes t^m,Y\otimes t^n)\hat{k}.$

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$$\widehat{\mathfrak{𝔤}}=L\mathfrak{𝔤}\oplus \mathbb{ℂ}\hat{k}\oplus \mathbb{ℂ}d$$ 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.