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| In [[mathematics]], an '''affine Lie algebra''' is an infinite-dimensional [[Lie algebra]] that is constructed in a canonical fashion out of a finite-dimensional [[simple Lie algebra]]. It is a [[Kac–Moody algebra]] for which the [[generalized Cartan matrix]] is positive semi-definite and has corank 1. From purely mathematical point of view, affine Lie algebras are interesting because their [[representation theory]], like representation theory of finite dimensional, [[semisimple Lie algebra]]s is much better understood than that of general Kac–Moody algebras. As observed by [[Victor Kac]], the [[Weyl-Kac character formula|character formula]] for representations of affine Lie algebras implies certain combinatorial identities, the [[Macdonald identities]].
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| Affine Lie algebras play an important role in [[string theory]] and [[conformal field theory]] due to the way they are constructed: starting from a simple Lie algebra <math>\mathfrak{g}</math>, one considers the '''loop algebra''', <math>L\mathfrak{g}</math>, formed by the <math>\mathfrak{g}</math>-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra <math>\hat{\mathfrak{g}}</math> is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physicists call a '''[[Anomaly (physics)|quantum anomaly]]''' and mathematicians a '''[[Group extension%23Central extension|central extension]]'''. More generally,
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| if σ is an [[automorphism]] of the simple Lie algebra <math>\mathfrak{g}</math> associated to an automorphism of its [[Dynkin diagram]], the '''twisted loop algebra''' <math>L_\sigma\mathfrak{g}</math> consists of <math>\mathfrak{g}</math>-valued functions ''f'' on the real line which satisfy
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| the twisted periodicity condition ''f(x+2π) = σ f(x)''. Their central extensions are precisely the '''twisted affine Lie algebras'''. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the [[modular group]].
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| == Affine Lie algebras from simple Lie algebras ==
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| === Definition ===
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| If <math>\mathfrak{g}</math> is a finite dimensional simple Lie algebra, the corresponding
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| affine Lie algebra <math>\hat{\mathfrak{g}}</math> is constructed as a [[Group extension%23Central extension|central extension]] of the infinite-dimensional Lie algebra <math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]</math>, with one-dimensional center <math>\mathbb{C}c.</math>
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| As a vector space, | |
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| : <math>\widehat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c,</math>
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| where <math>\mathbb{C}[t,t^{-1}]</math> is the complex vector space of [[Laurent series|Laurent polynomials]] in the indeterminate ''t''. The Lie bracket is defined by the formula
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| : <math>[a\otimes t^n+\alpha c, b\otimes t^m+\beta c]=[a,b]\otimes t^{n+m}+\langle a|b\rangle n\delta_{m+n,0}c</math>
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| for all <math>a,b\in\mathfrak{g}, \alpha,\beta\in\mathbb{C}</math> and <math>n,m\in\mathbb{Z}</math>, where <math>[a,b]</math> is the Lie bracket in the Lie algebra <math>\mathfrak{g}</math> and <math>\langle\cdot |\cdot\rangle</math> is the [[Killing form|Cartan-Killing form]] on <math>\mathfrak{g}.</math> | |
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| The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. There is a distinguished derivation of the affine Lie algebra defined by
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| : <math> \delta (a\otimes t^m+\alpha c) = t{d\over dt} (a\otimes t^m).</math>
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| The corresponding '''affine Kac-Moody algebra''' is defined by adding an extra generator ''d'' satisfying ''[d,A] = δ(A)'' (a [[semidirect product]]).
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| ===Constructing the Dynkin diagrams===
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| The [[Dynkin diagram]] of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root. Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group of [[Outer automorphism group|outer automorphisms]] of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an '''untwisted''' affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to '''twisted''' affine Lie algebras.
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| {| class=wikitable width=660
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| |+ [[Dynkin diagram]]s for affine Lie algebras
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| |- align=center
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| |valign=top|[[File:Affine Dynkin diagrams.png|360px]]<BR>The set of extended (untwisted) affine Dynkin diagrams, with added nodes in green
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| |[[File:Twisted affine Dynkin diagrams.png|300px]]<BR>"Twisted" affine forms are named with (2) or (3) superscripts.<BR>(''k'' is the number of nodes in the graph)
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| |}
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| ===Classifying the central extensions===
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| The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as a [[Group extension%23Central extension|central extension]] of the loop algebra of the corresponding simple Lie algebra. If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra. In physics, one often considers instead the direct sum of a semisimple algebra and an abelian algebra <math>\mathbb{C}^n</math>. In this case one also needs to add ''n'' further central elements for the ''n'' abelian generators.
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| The second integral cohomology of the loop group of the corresponding simple compact Lie group is isomorphic to the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the first [[Chern class]] of the [[fibration]]. Therefore the central extensions of an affine Lie group are classified by a single parameter ''k'' which is called the ''level'' in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist when ''k'' is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge for each simple component.
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| ==Applications==
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| They appear naturally in [[theoretical physics]] (for example, in [[conformal field theory|conformal field theories]] such as the [[WZW model]] and [[coset model]]s and even on the worldsheet of the [[heterotic string]]), geometry, and elsewhere in mathematics.
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| ==References==
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| * {{citation|first=P.|last= Di Francesco|first2=P. |last2=Mathieu|first3=D. |last3=Sénéchal|title=Conformal Field Theory''|publisher=Springer-Verlag| year=1997|isbn=0-387-94785-X}}
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| *{{citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|isbn=0-521-48412-X}}
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| *{{citation|first=Peter|last=Goddard|authorlink=Peter Goddard (physicist)|first2=David|last2=Olive|title=Kac-Moody and Virasoro algebras: A Reprint Volume for Physicists|series=Advanced Series in Mathematical Physics|volume=3|publisher=World Scientific|year=1988|isbn=9971-5-0419-7}}
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| *{{citation|first=Victor|last= Kac|authorlink=Victor Kac|title=Infinite dimensional Lie algebras |edition=3|publisher=Cambridge University Press|year= 1990|isbn=0-521-46693-8}}
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| *{{citation|first=Toshitake|last= Kohno|title=Conformal Field Theory and Topology|year=1998|publisher=American Mathematical Society|isbn=0-8218-2130-X}}
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| *{{citation|first1=Andrew|last1=Pressley|first2=Graeme|last2=Segal|authorlink2=Graeme Segal|title=Loop groups|publisher=Oxford University Press|year=1986|isbn=0-19-853535-X}}
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| [[Category:Lie algebras]]
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| [[Category:Conformal field theory]]
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