affine Lie algebras

E440255

Kac–Moody algebra infinite-dimensional Lie algebra mathematical concept

Affine Lie algebras are infinite-dimensional extensions of finite-dimensional simple Lie algebras that play a central role in representation theory, conformal field theory, and the study of exactly solvable models in mathematical physics.

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Predicate	Object
instanceOf	Kac–Moody algebra ⓘ infinite-dimensional Lie algebra ⓘ mathematical concept ⓘ
appearsIn	Frenkel–Lepowsky–Meurman’s work on the Monster vertex algebra ⓘ Kac’s book Infinite Dimensional Lie Algebras NERFINISHED ⓘ
classificationBasedOn	affine Dynkin diagrams ⓘ generalized Cartan matrices of affine type ⓘ
definedAs	Kac–Moody algebras whose generalized Cartan matrix is of affine type ⓘ central extensions of loop algebras of finite-dimensional simple Lie algebras with derivation ⓘ
hasExample	affine Lie algebra of type A_2^(2) ⓘ affine Lie algebra of type A_n^(1) ⓘ affine Lie algebra of type D_4^(3) ⓘ affine Lie algebra of type D_n^(1) ⓘ affine Lie algebra of type E_8^(1) ⓘ twisted affine Lie algebras ⓘ untwisted affine Lie algebras ⓘ
hasInvariant	Cartan matrix of affine type ⓘ affine Weyl group NERFINISHED ⓘ central charge in associated conformal field theory ⓘ level of a representation ⓘ
hasKeyContributor	Anthony Joseph NERFINISHED ⓘ George Lusztig NERFINISHED ⓘ Igor Frenkel NERFINISHED ⓘ James Lepowsky NERFINISHED ⓘ Masaki Kashiwara NERFINISHED ⓘ Robert Moody NERFINISHED ⓘ Victor Kac NERFINISHED ⓘ
hasProperty	admit Cartan subalgebras ⓘ admit Chevalley generators and relations ⓘ admit integrable highest weight representations ⓘ admit level decomposition of representations ⓘ admit triangular decomposition ⓘ are usually defined over the complex numbers ⓘ central extension of loop algebras ⓘ graded by the integers ⓘ have Weyl groups that are affine Weyl groups ⓘ have degree derivation ⓘ have generalized Cartan matrix of affine type ⓘ have imaginary roots ⓘ have one-dimensional center ⓘ have real roots ⓘ have root systems of affine type ⓘ
hasStructure	Borel subalgebras NERFINISHED ⓘ Cartan subalgebra ⓘ root space decomposition ⓘ triangular decomposition into positive, Cartan, and negative parts ⓘ
playsRoleIn	Knizhnik–Zamolodchikov equations NERFINISHED ⓘ Sugawara construction of the Virasoro algebra NERFINISHED ⓘ construction of the Monster module ⓘ proofs of modular invariance of characters ⓘ
relatedTo	Drinfeld–Jimbo quantum affine algebras NERFINISHED ⓘ Kac–Moody algebras NERFINISHED ⓘ Wess–Zumino–Witten models NERFINISHED ⓘ Yangians NERFINISHED ⓘ affine Weyl groups NERFINISHED ⓘ affine root systems ⓘ conformal field theory ⓘ current algebras ⓘ exactly solvable models in statistical mechanics ⓘ finite-dimensional simple Lie algebras ⓘ integrable systems ⓘ loop algebras ⓘ modular forms ⓘ quantum groups ⓘ representation theory ⓘ theta functions ⓘ vertex operator algebras ⓘ
subclassOf	Kac–Moody algebras NERFINISHED ⓘ infinite-dimensional complex Lie algebras ⓘ symmetrizable Kac–Moody algebras ⓘ
usedIn	construction of vertex operator algebras ⓘ string theory ⓘ theory of exactly solvable lattice models ⓘ theory of integrable quantum field theories ⓘ theory of modular tensor categories ⓘ two-dimensional conformal field theory ⓘ

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Rogers–Ramanujan-type identities → relatedTo → affine Lie algebras ⓘ