Affine Lie algebra

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 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 conformal field theory due to the way they are constructed: starting from a simple Lie algebra ${\displaystyle {\mathfrak {g}}}$, one considers the loop algebra, ${\displaystyle L{\mathfrak {g}}}$, formed by the ${\displaystyle {\mathfrak {g}}}$-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra ${\displaystyle {\hat {\mathfrak {g}}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ associated to an automorphism of its Dynkin diagram, the twisted loop algebra ${\displaystyle L_{\sigma }{\mathfrak {g}}}$ consists of ${\displaystyle {\mathfrak {g}}}$-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.

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