# 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.