Quasi-finite modules over affine and extended affine Lie algebras
Souvik Pal
TL;DR
This work classifies irreducible quasi-finite (weakly integrable) modules with non-trivial core action over untwisted finitely generated center (fgc) extended affine Lie algebras, focusing on nullities 1 and 2; it shows such modules are either integrable with trivial core action or restricted generalized highest weight modules, and in nullity 2 these restricted GHWs are highest weight type modules. The authors develop a unified framework via Lie tori, multiloop realizations, and derivation-centered central extensions to realize and classify these modules, extending Chari–Pressley and Rao–Futorny results to higher-nullity EALAs such as toroidal and minimal EALAs. They provide complete classifications for the affine case and for nullity-2 fgc EALAs, and give structural descriptions (e.g., highest weight type and restricted GHW modules) that generalize to toroidal and Hamiltonian-like families. The results illuminate the representation theory of higher-nullity EALAs, with implications for toroidal algebras and related structures, and offer a concrete pathway toward classifying irreducible quasi-finite modules across broader EALA families.
Abstract
In this paper, we consider irreducible quasi-finite (or equivalently weakly integrable) modules, with non-trivial action of the core, over the extended affine Lie algebras (EALAs) whose centerless cores are multiloop algebras. The centerless cores of all but one family of EALAs having nullity greater than 1 are known to admit such multiloop realizations. For any such (untwisted) EALA, we show that the irreducible quasi-finite modules are either integrable with the center of the underlying core acting trivially, or restricted generalized highest weight (GHW) modules. We further prove that in the nullity 2 case, these irreducible restricted GHW modules turn out to be highest weight type modules, thereby classifying the irreducible quasi-finite modules over all such EALAs. In particular, we obtain the classification of irreducible quasi-finite modules over toroidal Lie algebras, minimal EALAs and toroidal EALAs of nullity 2. Along the way, we completely classify the irreducible weakly integrable modules over affine Kac-Moody algebras (studied by Rao-Futorny [Trans. Amer. Math. Soc. 2009] for non-zero level modules). Our results generalize the well-known work of Chari [Invent. Math. 1986] and Chari-Pressley [Math. Ann. 1986] concerning the classification of irreducible integrable modules over (nullity 1) affine Kac-Moody algebras.