In this paper, we classify the irreducible integrable modules for nullity 2 toroidal extended affine Lie algebras with finite dimensional weight spaces.

We generalize Feigin and Miwa’s construction of extended vertex operator (super)algebras Ak(sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type.

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras have been under intensive investigation in recent years. They have also been call...

Motivation. The theory of affine (Kac-Moody) Lie algebras has been a tremendous success story. Not only has one been able to generalize essentially all of the well-developed theory of finite-dimensional simple Lie algebras and their associated groups to the setting of affine Lie algebras, but these algebras have found many striking applications in other parts of mathematics. It is natural to as...