Quasi-integrable modules (Critical level)

Asghar Daneshvar, Hajar Kiamehr, Malihe Yousofzadeh

Published 2024-02-26, updated 2025-02-13Version 2

Representation theory of Lie (super)algebras has garnered significant research attention over many years, specially due to its applications in theoretical physics; in this regard, representation theory of affine Lie (super)algebras is of great importance. To characterize simple modules over affine Lie (super)algebras, we need to address the cases of nonzero and critical level separately. Although a vast amount of research exists on the representation theory of affine Lie (super)algebras, investigations regarding general modules at the critical level are few. In the current paper, we focus on twisted affine Lie superalgebras {$\LL$} ($\LL_1\neq \{0\}$). The even part of almost all such {$\LL$}'s contains two affine Lie algebras and in turn, the behavior of the restricted modules corresponding to these affine components and their interactions play important roles in characterizing simple $\LL$-modules. In this paper, we deal with $\LL$-modules, at the critical level, for which only one of the restricted modules corresponding to the affine components of {$\LL_0$}, is integrable. The results are important from two perspectives: first, we provide a characterization of a class of modules at the critical level and second, we will use this to characterize general modules at the critical level.