Borel Subalgebras and Categories of Highest Weight Modules for Toroidal Lie Algebras

In this article we begin an investigation of the conjugacy classes of Borel subalgebras together with Verma modules induced from “standard” Borel subalgebras of a toroidal Lie algebra in two variables. We define, for each highest weight λ, a category λ of representations of that contain these Verma modules and show that when a certain central element acts nontrivially this category is equivalent to the category λ( ) for an extension of a suitable affine Kac–Moody algebra . From this equivalence we obtain a BGG type resolution and BGG duality theorem in the setting of λ.