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 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 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.
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In algebra, a simple Liealgebra is a Liealgebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras...
Affinealgebra may refer to: AffineLiealgebra, a type of Kac–Moody algebras The Liealgebra of the affine group Finitely-generated algebraAffine Hecke...
quantum affinealgebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affineLiealgebra. They...
In mathematics, loop algebras are certain types of Liealgebras, of particular interest in theoretical physics. For a Liealgebra g {\displaystyle {\mathfrak...
conformal field theories, whose symmetry is described by an abelian affineLiealgebra. Since they are free i.e. non-interacting, free bosonic CFTs are easily...
Note that every complex Liealgebra can also be viewed as a real Liealgebra of twice the dimension. The Liealgebra of affine transformations of dimension...
In mathematics, the special linear Liealgebra of order n over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak {sl}}_{n}F} or...
of representation theory, a Liealgebra representation or representation of a Liealgebra is a way of writing a Liealgebra as a set of matrices (or endomorphisms...
In algebraic geometry, an affinealgebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the...
(P, η) defines the structure of an affine geometry on M, making it into an affine manifold. The affineLiealgebra aff(n) splits as a semidirect product...
Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Liealgebra and the unique central extension of the Witt algebra. It is...
E_{8},F_{4},G_{2}} ), while affine type indecomposable matrices classify the affineLiealgebras (say over some algebraically closed field of characteristic...
Dynkin diagrams arise in the classification of semisimple Liealgebras over algebraically closed fields, in the classification of Weyl groups and other...
used to read off the list of simple Liealgebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle...
way as representations of semisimple Liealgebras. AffineLiealgebras are a special case of Kac–Moody algebras, which have particular importance in mathematics...
field of Lie theory, there are two definitions of a compact Liealgebra. Extrinsically and topologically, a compact Liealgebra is the Liealgebra of a compact...
mathematics, a Liealgebra is semisimple if it is a direct sum of simple Liealgebras. (A simple Liealgebra is a non-abelian Liealgebra without any non-zero...
ring). *-algebraAffineLiealgebra Akivis algebraAlgebra for a monad Albert algebra Alternative algebra AW*-algebra Azumaya algebra Banach algebra Birman–Wenzl...
important class of algebraic groups is given by the affinealgebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly...
In mathematics, an affinealgebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in...
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