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Concept
Affine Lie algebra
Summary
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. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. 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 , one considers the loop algebra, , formed by the -valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra is obtained by adding one extra dimension to the loop algebra and modifying a 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 consists of -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.
If is a finite-dimensional simple Lie algebra, the corresponding
affine Lie algebra is constructed as a central extension of the loop algebra , with one-dimensional center
As a vector space,
where is the complex vector space of Laurent polynomials in the indeterminate t.
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Related concepts (6)
Lie algebra extension
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.
Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. The Virasoro algebra is spanned by generators Ln for n ∈ Z and the central charge c. These generators satisfy and The factor of is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra.
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).