Classification of irreducible integrable representations of loop toroidal Lie algebras

The aim of this paper is to classify irreducible integrable representations of loop toroidal Lie algebras with finite dimensional weight spaces. Loop toroidal Lie algebras are given by [Formula: see text], where [Formula: see text] is a toroidal Lie algebra, [Formula: see text] is a commutative, associative, finitely generated unital algebra over [Formula: see text] and [Formula: see text] is the space spanned by degree derivations [Formula: see text] over [Formula: see text]. When the center of toroidal Lie algebra acts non-trivially on modules, we prove that modules come from direct sum of finitely many copies of affine Kac–Moody Lie algebras. In the case when the center of toroidal Lie algebra acts trivially on modules, we prove modules come from direct sum of finitely many copies of finite dimensional simple Lie algebras.