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

Definition

For a Lie algebra ${\displaystyle {\mathfrak {g))}$ over a field ${\displaystyle K}$, if ${\displaystyle K[t,t^{-1}]}$ is the space of Laurent polynomials, then

$${\displaystyle L{\mathfrak {g)):={\mathfrak {g))\otimes K[t,t^{-1}],}$$

$${\displaystyle [X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}.}$$

Geometric definition

If ${\displaystyle {\mathfrak {g))}$ is a Lie algebra, the tensor product of ${\displaystyle {\mathfrak {g))}$ with C^∞(S¹), the algebra of (complex) smooth functions over the circle manifold S¹ (equivalently, smooth complex-valued periodic functions of a given period),

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

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

$${\displaystyle [g_{1}\otimes f_{1},g_{2}\otimes f_{2}]=[g_{1},g_{2}]\otimes f_{1}f_{2}.}$$

Here g₁ and g₂ are elements of ${\displaystyle {\mathfrak {g))}$ and f₁ and f₂ are elements of C^∞(S¹).

This isn't precisely what would correspond to the direct product of infinitely many copies of ${\displaystyle {\mathfrak {g))}$, 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 ${\displaystyle {\mathfrak {g))}$; a smooth parametrized loop in ${\displaystyle {\mathfrak {g))}$, in other words. This is why it is called the loop algebra.

Gradation

Defining ${\displaystyle {\mathfrak {g))_{i))$ to be the linear subspace ${\displaystyle {\mathfrak {g))_{i}={\mathfrak {g))\otimes t^{i}<L{\mathfrak {g)),}$ the bracket restricts to a product

$${\displaystyle [\cdot \,,\,\cdot ]:{\mathfrak {g))_{i}\times {\mathfrak {g))_{j}\rightarrow {\mathfrak {g))_{i+j},}$$

${\displaystyle \mathbb {Z} }$

In particular, the bracket restricts to the 'zero-mode' subalgebra ${\displaystyle {\mathfrak {g))_{0}\cong {\mathfrak {g))}$.

Derivation

There is a natural derivation on the loop algebra, conventionally denoted ${\displaystyle d}$ acting as

$${\displaystyle d:L{\mathfrak {g))\rightarrow L{\mathfrak {g))}$$

$${\displaystyle d(X\otimes t^{n})=nX\otimes t^{n))$$

${\displaystyle d=t{\frac {d}{dt))}$

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 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.

Affine Lie algebras as central extension of loop algebras

If ${\displaystyle {\mathfrak {g))}$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra ${\displaystyle L{\mathfrak {g))}$ gives rise to an affine Lie algebra. Furthermore this central extension is unique.^[1]

The central extension is given by adjoining a central element ${\displaystyle {\hat {k))}$, that is, for all ${\displaystyle X\otimes t^{n}\in L{\mathfrak {g))}$,

$${\displaystyle [{\hat {k)),X\otimes t^{n}]=0,}$$

$${\displaystyle [X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}+mB(X,Y)\delta _{m+n,0}{\hat {k)),}$$

${\displaystyle B(\cdot ,\cdot )}$

The central extension is, as a vector space, ${\displaystyle L{\mathfrak {g))\oplus \mathbb {C} {\hat {k))}$ (in its usual definition, as more generally, ${\displaystyle \mathbb {C} }$ 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

$${\displaystyle \varphi :L{\mathfrak {g))\times L{\mathfrak {g))\rightarrow \mathbb {C} }$$

$${\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n})=mB(X,Y)\delta _{m+n,0}.}$$

${\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n}){\hat {k)).}$

Affine Lie algebra

In physics, the central extension ${\displaystyle L{\mathfrak {g))\oplus \mathbb {C} {\hat {k))}$ is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space^[2]

$${\displaystyle {\hat {\mathfrak {g))}=L{\mathfrak {g))\oplus \mathbb {C} {\hat {k))\oplus \mathbb {C} d}$$

${\displaystyle d}$

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.