Representations of toroidal and full toroidal Lie algebras over polynomial algebras

Toroidal Lie algebras are n variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of n-variable generalization of Affine-Virasoro algebras. Let (h) over tilde be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on U((h) over tilde) for all toroidal Lie algebras as well as full toroidal Lie algebras which are free U((h) over tilde)-modules of rank 1. These modules exist only for type A(l)(l >= 1), C-l(l >= 2) toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.