Simple affine vertex algebras at admissible levels are semi-simple in the category O, but beyond the category O they contain interesting categories of representations with many new research challenges. We will first present our explicit lattice realizations of simple affine VOA Lk(𝔰𝔩2) at arbitrary admissible level k, and their modules in certain categories. Then we discuss the existence and explicit realization of logarithmic modules which appear as extensions of weight modules. The next natural task is to include Whittaker modules in the representation category. Although Whittaker modules are constructed using standard Lie-theoretic constructions, we will show that in order to understand the structure of affine Whittaker modules, one needs to apply vertex-algebraic techniques. We present explicit realization of Whittaker modules for some vertex algebras. We will discuss our recent efforts to generalize this realization in higher-rank cases.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
