2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132966We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.Representation Theory17B66; 17B69text