φ-imaginary Verma modules and their generalizations for the toroidal Lie algebras

Abstract

In this paper, we first construct a class of weight modules, called φ-imaginary Verma modules, for the toroidal Lie algebras. Then a criterion for the irreducibility of the φ-imaginary Verma modules is obtained and the irreducible quotients for the reducible ones are studied. Furthermore, we construct a more general class of ℤn-graded modules for the toroidal Lie algebras and we discuss their irreducibility. This class of modules includes the φ-imaginary Verma modules as special examples.