Open AccessSpinor representations of affine Lie algebras
Author(s) -
Igor Frenkel
Publication year - 1980
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.77.11.6303
Subject(s) - spinor , pure mathematics , mathematics , fundamental representation , lie algebra , affine lie algebra , type (biology) , killing form , affine transformation , connection (principal bundle) , kac–moody algebra , dynkin diagram , irreducible representation , representation theory of su , algebra over a field , lie conformal algebra , adjoint representation of a lie algebra , mathematical physics , current algebra , weight , ecology , geometry , biology
Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the typesD l +1(2) ,B l (1) , orD l (1) . These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of typesBl orDl of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the typeD l +1(2) , three forB l (1) , and four forD l (1) . We note the profound connection of our construction with quantum field theory—in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of typesA l (1) ,D l (1) ,E l (1) , we obtain classical Jacobi identities and addition formulas for elliptic θ-functions.