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- Affine_Lie_algebra abstract "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 , 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 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 satisfythe 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.".
- Affine_Lie_algebra wikiPageID "1350865".
- Affine_Lie_algebra wikiPageRevisionID "584043625".
- Affine_Lie_algebra hasPhotoCollection Affine_Lie_algebra.
- Affine_Lie_algebra subject Category:Conformal_field_theory.
- Affine_Lie_algebra subject Category:Lie_algebras.
- Affine_Lie_algebra type Abstraction100002137.
- Affine_Lie_algebra type Algebra106012726.
- Affine_Lie_algebra type Cognition100023271.
- Affine_Lie_algebra type Content105809192.
- Affine_Lie_algebra type Discipline105996646.
- Affine_Lie_algebra type KnowledgeDomain105999266.
- Affine_Lie_algebra type LieAlgebras.
- Affine_Lie_algebra type Mathematics106000644.
- Affine_Lie_algebra type PsychologicalFeature100023100.
- Affine_Lie_algebra type PureMathematics106003682.
- Affine_Lie_algebra type Science105999797.
- Affine_Lie_algebra comment "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.".
- Affine_Lie_algebra label "Affine Lie algebra".
- Affine_Lie_algebra label "Affine Lie-Algebra".
- Affine_Lie_algebra sameAs Affine_Lie-Algebra.
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- Affine_Lie_algebra sameAs Q632521.
- Affine_Lie_algebra sameAs Q632521.
- Affine_Lie_algebra sameAs Affine_Lie_algebra.
- Affine_Lie_algebra wasDerivedFrom Affine_Lie_algebra?oldid=584043625.
- Affine_Lie_algebra isPrimaryTopicOf Affine_Lie_algebra.