As observed by Victor 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 and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra
-valued functions on a circle (interpreted as the closed string) with pointwise commutator.
is obtained by adding one extra dimension to the loop algebra and modifying the commutator in a non-trivial way, which physicists call a quantum anomaly (in this case, the anomaly of the WZW model) and mathematicians a central extension.
More generally, if σ is an automorphism of the simple Lie algebra
associated to an automorphism of its Dynkin diagram, the twisted loop algebra
-valued functions f on the real line which satisfy 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.
is the complex vector space of Laurent polynomials in the indeterminate t. The Lie bracket is defined by the formula for all
There is a distinguished derivation of the affine Lie algebra defined by The corresponding affine Kac–Moody algebra is defined as a semidirect product by adding an extra generator d that satisfies [d, A] = δ(A).
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 to twisted affine Lie algebras.
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 this case one also needs to add n further central elements for the n abelian generators.
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.
Fix a finite-dimensional, simple, complex Lie algebra
The Killing form can almost be completely determined using its invariance property.
The representation theory for affine Lie algebras is usually developed using Verma modules.
Just as in the case of semi-simple Lie algebras, these are highest weight modules.
Roughly speaking, this follows because the Killing form is Lorentzian in the
The vacuum representation in fact can be equipped with vertex algebra structure, in which case it is called the affine vertex algebra of rank
The usual denominator identities of semi-simple Lie algebras generalize as well; because the characters can be written as "deformations" or q-analogs of the highest weights, this led to many new combinatoric identities, include many previously unknown identities for the Dedekind eta function.
This allows affine Lie algebras to serve as symmetry algebras of conformal field theories such as WZW models or coset models.
As a consequence, affine Lie algebras also appear in the worldsheet description of string theory.