Toroidal Z-Algebras

Abstract

The Toroidal Lie algebras are n variable genaralizations of Affine Kac-Moody Lie algebras. As in the affine Lie algebras there exists finite order auto= morphisms corresponding to Dynkin diagram automorphisms. The fixed point sub= algebras are called twisted toroidal Lie algebras. In this paper we construt faithfull representations for the twisted toroidal Lie algebras (this includes the non-twisted case also) useing methods developed by Lepowsky-Wilson [LW]. This construction recovers the result by Eswara Rao-Moody [EM] in the homogeneous picture and by Yuly Billig [B1] in the principal picture. The proofs given in this paper are much shorter than the above works. The results for the twisted case are compleetly new.