Editing Loop algebra (section)

==Affine Lie algebras as central extension of loop algebras==
{{See also |Lie algebra extension#Polynomial loop-algebra |Affine Lie algebra}}
If <math>\mathfrak{g}</math> is a [[semisimple Lie algebra]], then a nontrivial [[Group extension#Central extension|central extension]] of its loop algebra <math>L\mathfrak g</math> gives rise to an [[affine Lie algebra]]. Furthermore this central extension is unique.<ref>{{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.}}</ref>

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

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

=== Cocycle ===
{{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<math display=block>\varphi: L\mathfrak g \times L\mathfrak g \rightarrow \mathbb{C}</math>
satisfying
<math display=block>\varphi(X\otimes t^m, Y\otimes t^n) = mB(X,Y)\delta_{m+n,0}.</math>
Then the extra term added to the bracket is <math>\varphi(X\otimes t^m, Y\otimes t^n)\hat k.</math>

===Affine Lie algebra===
In physics, the central extension <math>L\mathfrak g \oplus \mathbb C \hat k</math> is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space<ref name="BYB">P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, {{ISBN|0-387-94785-X}}</ref><math display=block>\hat \mathfrak{g} = L\mathfrak{g} \oplus \mathbb C \hat k \oplus \mathbb C d</math>
where <math>d</math> 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.