loop algebra

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

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

with the inherited bracket

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

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

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

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

$$[g\_1\backslash otimes\; f\_1,g\_2\; \backslash otimes\; f\_2]=[g\_1,g\_2]\backslash otimes\; f\_1\; f\_2.$$

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

This isn't precisely what would correspond to the direct product of infinitely many copies of $\backslash mathfrak\{g\}$, one for each point in {{math|S¹}}, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from {{math|S¹}} to $\backslash mathfrak\{g\}$; a smooth parametrized loop in $\backslash mathfrak\{g\}$, in other words. This is why it is called the loop algebra.

Defining $\backslash mathfrak\{g\}\_i$ to be the linear subspace the bracket restricts to a product$$[\backslash cdot\backslash ,\; ,\; \backslash ,\; \backslash cdot]:\; \backslash mathfrak\{g\}\_i\; \backslash times\; \backslash mathfrak\{g\}\_j\; \backslash rightarrow\; \backslash mathfrak\{g\}\_\{i+j\},$$

hence giving the loop algebra a $\backslash mathbb\{Z\}$-graded Lie algebra structure.

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

{{See also|Derivation (differential algebra)}}

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

$$d:\; L\backslash mathfrak\{g\}\; \backslash rightarrow\; L\backslash mathfrak\{g\}$$

$$d(X\backslash otimes\; t^n)\; =\; nX\backslash otimes\; t^n$$

and so can be thought of formally as $d\; =\; t\backslash frac\{d\}\{dt\}$.

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 {{math|S¹}} to a Lie group {{math|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.

{{See also |Lie algebra extension#Polynomial loop-algebra |Affine Lie algebra}}

If $\backslash mathfrak\{g\}$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra $L\backslash mathfrak\; g$ gives rise to an affine Lie algebra. Furthermore this central extension is unique.{{cite book |first=V.G. |last=Kac|title=Infinite-dimensional Lie algebras|edition=3rd|publisher=Cambridge University Press|year=1990|author-link=Victor Kac|isbn=978-0-521-37215-2 |at=Exercise 7.8.}}

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

$$[\backslash hat\; k,\; X\backslash otimes\; t^n]\; =\; 0,$$

and modifying the bracket on the loop algebra to

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

where $B(\backslash cdot,\; \backslash cdot)$ is the Killing form.

The central extension is, as a vector space, $L\backslash mathfrak\{g\}\; \backslash oplus\; \backslash mathbb\{C\}\backslash hat\; k$ (in its usual definition, as more generally, $\backslash mathbb\{C\}$ can be taken to be an arbitrary field).

{{See also|Lie algebra extension#Central}}

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$$\backslash varphi:\; L\backslash mathfrak\; g\; \backslash times\; L\backslash mathfrak\; g\; \backslash rightarrow\; \backslash mathbb\{C\}$$

satisfying

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

Then the extra term added to the bracket is $\backslash varphi(X\backslash otimes\; t^m,\; Y\backslash otimes\; t^n)\backslash hat\; k.$

In physics, the central extension $L\backslash mathfrak\; g\; \backslash oplus\; \backslash mathbb\; C\; \backslash hat\; k$ is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector spaceP. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, {{ISBN|0-387-94785-X}}$$\backslash hat\; \backslash mathfrak\{g\}\; =\; L\backslash mathfrak\{g\}\; \backslash oplus\; \backslash mathbb\; C\; \backslash hat\; k\; \backslash oplus\; \backslash mathbb\; C\; d$$

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.

{{reflist}}

{{refbegin}}

• {{citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|isbn=0-521-48412-X}}

{{refend}}

{{String theory topics |state=collapsed}}

Category:Lie algebras

Category:String theory

Category:Conformal field theory