Z-algebras and parafermion algebras

Abstract

We continue our discussion in Chapter 13, but we shall relativize it to certain subspaces h _* of h, following the setting of Chapters 2, 3 and 4. Recall that G is an affine Lie algebra of type Â, D or Ê and that l is a positive integer. Restricting our attention to the vacuum space of L(l,0) with respect to a natural Heisenberg subalgebra of G, and passing to the quotient spaces of this vacuum space defined by the actions of certain infinite abelian groups, we shall construct canonical generalized vertex operator algebras. Moreover, the corresponding quotient spaces for the vacuum space of any level l standard G-module (which can be obtained from a tensor product of basic modules) are modules for these generalized vertex operator algebras. If we make a special choice of the infinite abelian group, the algebra turns out to be simple and the modules turn out to be irreducible. As an illustration, we specialize our construction to the case of the affine Lie algebra A₁⁽¹⁾ , and in this case we show in detail the essential equivalence between Z-algebras ([LP1]-[LP3]) and parafermion algebras [ZF1] by realizing the parafermion algebras as canonically modified Z-algebras acting on certain quotient spaces of the vacuum spaces of standard A₁⁽¹⁾ -modules defined by the action of an infinite cyclic group (a construction carried out in [LP1]).