Spinor representations of affine Lie algebras

Abstract

Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types D(l+1) ((2)), B(l) ((1)), or D(l) ((1)). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types B(l) or D(l) 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 type D(l+1) ((2)), three for B(l) ((1)), and four for D(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 types A(l) ((1)), D(l) ((1)), E(l) ((1)), we obtain classical Jacobi identities and addition formulas for elliptic theta-functions.