Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types Dl+1(2), Bl(1), or Dl(1). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types Bl or Dl 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 Dl+1(2), three for Bl(1), and four for Dl(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 Al(1), Dl(1), El(1), we obtain classical Jacobi identities and addition formulas for elliptic θ-functions.
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机译：令[unk]为Dl + 1 （2），Bl （1）或Dl 类型之一的无穷维Kac-Moody Lie代数（1）。这些代数的特征在于以下特性：消除它们的Dynkin图的任何端点，即可得到经典正交李代数的Bl或Dl类型的图。根据经典情况的类比，我们构造李代数[unk]的两个表示形式，我们称其为旋子表示形式。我们获得了每个自旋表示都是不可约的或具有两个不可约的分量。这为我们提供了[unk]基本表示的显式构造，两个用于类型Dl + 1 （2），三个用于Bl （1），四个用于Dl （1）。我们注意到我们的构造与量子场论（特别是与费米子场）的深远联系。将我们的表示的字符公式与Al （1）， D l 类型的Kac-Moody Lie代数的基本表示的另一种构造进行比较（1）， E l （1），我们获得了经典的Jacobi身份和椭圆θ-的加法公式。功能。
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