# Loop algebra

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{{Short description|Type of Lie algebra of interest in physics}}
{{distinguish|text=the term for [quasigroups with an identity element](/source/Quasigroup), also called an algebraic loop}}

In [mathematics](/source/mathematics), '''loop algebras''' are certain types of [Lie algebra](/source/Lie_algebra)s, of particular interest in [theoretical physics](/source/theoretical_physics).

==Definition==
For a Lie algebra <math>\mathfrak{g}</math> over a field <math>K</math>, if <math>K[t,t^{-1}]</math> is the space of [Laurent polynomials](/source/Laurent_polynomials), then
<math display=block>L\mathfrak{g} := \mathfrak{g}\otimes K[t,t^{-1}],</math>
with the inherited bracket
<math display=block>[X\otimes t^m, Y\otimes t^n] = [X,Y]\otimes t^{m+n}.</math>

=== Geometric definition ===

If <math>\mathfrak{g}</math> is a Lie algebra, the [tensor product](/source/tensor_product) of <math>\mathfrak{g}</math> with {{math|''C''<sup>∞</sup>(''S''<sup>1</sup>)}}, the [algebra](/source/associative_algebra) of (complex) [smooth function](/source/smooth_function)s over the [circle](/source/n-sphere) [manifold](/source/manifold) {{math|''S''<sup>1</sup>}} (equivalently, smooth complex-valued [periodic function](/source/periodic_function)s of a given period),

<math display=block>\mathfrak{g}\otimes C^\infty(S^1),</math>

is an infinite-dimensional Lie algebra with the [Lie bracket](/source/Lie_bracket_of_vector_fields) given by

<math display=block>[g_1\otimes f_1,g_2 \otimes f_2]=[g_1,g_2]\otimes f_1 f_2.</math>

Here {{math|''g''<sub>1</sub>}} and {{math|''g''<sub>2</sub>}} are elements of <math>\mathfrak{g}</math> and {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}} are elements of {{math|''C''<sup>∞</sup>(''S''<sup>1</sup>)}}.

This isn't precisely what would correspond to the [direct product](/source/direct_product) of infinitely many copies of <math>\mathfrak{g}</math>, one for each point in {{math|''S''<sup>1</sup>}}, because of the smoothness restriction. Instead, it can be thought of in terms of [smooth map](/source/smooth_map) from {{math|''S''<sup>1</sup>}} to <math>\mathfrak{g}</math>; a smooth parametrized loop in <math>\mathfrak{g}</math>, in other words. This is why it is called the '''loop algebra'''.

== Gradation ==
Defining <math>\mathfrak{g}_i</math> to be the [linear subspace](/source/linear_subspace) <math>\mathfrak{g}_i = \mathfrak{g}\otimes t^i < L\mathfrak{g},</math> the bracket restricts to a product<math display=block>[\cdot\, , \, \cdot]: \mathfrak{g}_i \times \mathfrak{g}_j \rightarrow \mathfrak{g}_{i+j},</math>
hence giving the loop algebra a <math>\mathbb{Z}</math>-[graded Lie algebra](/source/graded_Lie_algebra) structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra <math>\mathfrak{g}_0 \cong \mathfrak{g}</math>.

== Derivation ==
{{See also|Derivation (differential algebra)}}
There is a natural derivation on the loop algebra, conventionally denoted <math>d</math> acting as
<math display=block>d: L\mathfrak{g} \rightarrow L\mathfrak{g}</math>
<math display=block>d(X\otimes t^n) = nX\otimes t^n</math>
and so can be thought of formally as <math>d = t\frac{d}{dt}</math>.

It is required to define [affine Lie algebra](/source/affine_Lie_algebra)s, which are used in physics, particularly [conformal field theory](/source/conformal_field_theory).

==Loop group==
Similarly, a set of all smooth maps from {{math|''S''<sup>1</sup>}} to a [Lie group](/source/Lie_group) {{math|''G''}} forms an infinite-dimensional Lie group (Lie group in the sense we can define [functional derivative](/source/functional_derivative)s over it) called the '''[loop group](/source/loop_group)'''. The Lie algebra of a loop group is the corresponding loop algebra.

==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](/source/semisimple_Lie_algebra), then a nontrivial [central extension](/source/Group_extension) of its loop algebra <math>L\mathfrak g</math> gives rise to an [affine Lie algebra](/source/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](/source/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](/source/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](/source/Lie_algebra_cohomology), the central extension can be described using a 2-[cocycle](/source/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.

==References==
{{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

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Adapted from the Wikipedia article [Loop algebra](https://en.wikipedia.org/wiki/Loop_algebra) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Loop_algebra?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.