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 h̃ 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(h̃) for all toroidal Lie algebras as well as full toroidal Lie algebras which are free U(h̃)-modules of rank 1. These modules exist only for type Al(l ≥ 1), Cl(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.