- Affine Lie algebra
In

mathematics , an**affine Lie algebra**is an infinite-dimensionalLie algebra that is constructed in a canonical fashion out of a finite-dimensionalsimple Lie algebra . It is aKac–Moody algebra whosegeneralized Cartan matrix is positive semi-definite and has corank 1. From purely mathematical point of view, affine Lie algebras are interesting because theirrepresentation theory , like representation theory of finite dimensional,semisimple Lie algebra s is much better understood than that of general (hyperbolic) Kac–Moody algebras. As observed byVictor Kac , the character formula for representations of affine Lie algebras implies certain combinatorial identities, the**Macdonald identities**.Affine Lie algebras play an important role in

string theory andconformal field theory due to the way they are constructed: starting from a simple Lie algebra $mathfrak\{g\}$, one considers the**loop algebra**, $Lmathfrak\{g\}$, formed by the $mathfrak\{g\}$-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra $hat\{mathfrak\{g$ is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physists call a**quantum anomaly**. 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 are given bymodular form s.**Affine Lie algebras from simple Lie algebras****Construction**Each (untwisted) affine Lie algebra may be constructed from a finite-dimensional semi-simple Lie algebra. Semi-simple Lie algebras are direct sums of commuting simple Lie algebras, and the corresponding affine Lie algebras are also direct sums of the affine versions of each simple Lie algebra. Thus it will suffice to consider affine Lie algebras constructed from simple Lie algebras.

The way in which we construct the affine Lie algebra $hat\{mathfrak\{g$ associated to a finite dimensional simple Lie algebra $mathfrak\{g\}$ goes as follows.

$hat\{mathfrak\{g$ is defined to be the central extension

$mathfrak\{g\}otimesmathbb\{C\}\; [t,t^\{-1\}]\; oplusmathbb\{C\}c$Where $mathbb\{C\}\; [t,t^\{-1\}]$ denotes the set of all Laurent polynomials in the indeterminate t. The bracket is defined by

$[aotimes\; t^n,\; botimes\; t^m]\; =\; [a,b]\; otimes\; t^nt^m+langle\; a|b\; angle\; ndelta\_\{m+n,0\}c$for all $a,binmathfrak\{g\}$ and $n,minmathbb\{Z\}$, where $langlecdot\; |cdot\; angle$ is the usual Cartan-Killing form on $mathfrak\{g\}$ and $c$ is the basis element of the central extension.

Note that the bracket on the left side of the definition is the "new" bracket being defined on $ilde\{mathfrak\{g$ and the bracket on the right side is the "old" bracket of $mathfrak\{g\}$. The algebra $mathfrak\{g\}otimesmathbb\{C\}\; [t,t^\{-1\}]$ is sometimes called the loop algebra associated to $mathfrak\{g\}$.

**Constructing the Dynkin diagrams**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 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 totwisted affine Lie algebra s.**Classifying the central extensions**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 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 $mathbb\{C\}^n$. In this case one also needs to add "n" further central elements for the "n" abelian generators.

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 thefibration . Therefore the central extensions of an affine Lie group are classified by a single parameter "k" which is called thecentral charge 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.**Noncompact real forms**Of course the number of central charges does not depend on the

real form that is chosen, as central charges are defined already for the complex Lie algebra, however different choices of real form lead to different unitary representations. If one chooses a noncompact real form one finds a much richer classification unitary representations, including some continuous families in which the central charge is not integral.**Applications**They appear naturally in

theoretical physics (for example, in conformal field theories such as theWZW model andcoset model s and even on the worldsheet of theheterotic string ), geometry, and elsewhere in mathematics.**References***

*citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|id= ISBN 0-521-48412-X

*citation|first=Peter|last=Goddard|authorlink=Peter Goddard|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|id=ISBN 9971-50-419-7

*citation|first=Victor|last= Kac|authorlink=Victor Kac|title=Infinite dimensional Lie algebras |edition=3|id= ISBN 0-521-46693-8|publisher=Cambridge University Press|year= 1990

*citation|first=Toshitake|last= Kohno|title=Conformal Field Theory and Topology|year=1998|publisher=American Mathematical Society|id= ISBN 0-8218-2130-X

*citation|first1=Andrew|last1=Pressley|first2=Graeme|last2=Segal|authorlink2=Graeme Segal|title=Loop groups|publisher=Oxford University Press|year=1986|id=ISBN 0-19-853535-X

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**En (Lie algebra)**— In mathematics, especially in Lie theory, E n is the Kac–Moody algebra whose Dynkin diagram is a line of n 1 points with an extra point attached to the third point from the end. Finite dimensional Lie algebras*E3 is another name for the Lie… … Wikipedia**Monster Lie algebra**— In mathematics, the monster Lie algebra is an infinite dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. Structure The monster Lie algebra m is a Z2 graded Lie… … Wikipedia**Affine algebra**— may refer to: * affine Lie algebra, a type of Kac–Moody algebras * the Lie algebra of the affine group * finitely generated algebra … Wikipedia**Affine connection**— An affine connection on the sphere rolls the affine tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the development. In the branch of mathematics called differential geometry, an… … Wikipedia**Lie group**— Lie groups … Wikipedia**Affine representation**— An affine representation of a topological (Lie) group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A , the affine group Aff( A ). Similarly, an affine representation of a Lie algebra g on… … Wikipedia**Affine space**— In mathematics, an affine space is an abstract structure that generalises the affine geometric properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one… … Wikipedia**Lie bracket of vector fields**— See Lie algebra for more on the definition of the Lie bracket and Lie derivative for the derivationIn the mathematical field of differential topology, the Lie bracket of vector fields or Jacobi ndash;Lie bracket is a bilinear differential… … Wikipedia**Affine curvature**— This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection. Special affine curvature, also known as the equi affine curvature or affine curvature, is a particular type of curvature… … Wikipedia**Affine action**— Let W be the Weyl group of a semisimple Lie algebra (associate to fixed choice of a Cartan subalgebra ). Assume that a set of simple roots in is chosen. The affine action (also called the dot action) of the Weyl group on the space is … Wikipedia