Quasi-finite modules over affine and extended affine Lie algebras

Title

Quasi-finite modules over affine and extended affine Lie algebras

Creator

Pal, Souvik

Description

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 also completely classify the irreducible weakly integrable modules over affine Kac-Moody algebras (RaoFutorny in Trans. Am. Math. Soc. 361(10): 54355455, 2009). Our results generalize the well-known work of Chari (Invent. Math. 85(2):317335, 1986) and ChariPressley (Math. Ann. 275(1):87104, 1986) concerning the classification of irreducible integrable modules over (nullity 1) affine KacMoody algebras. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2025.

Source

Mathematische Zeitschrift;Volume;311;Issue;3;Article No.;63;

Date

01-01-2025

Publisher

Springer Science and Business Media Deutschland GmbH

Subject

Extended affine Lie algebras; Fgc condition; Highest weight type modules; Quasi-finite modules; Weakly integrable modules

Coverage

Pal S., Department of Mathematics, Indian Institute of Science, CV Raman Road, Karnataka, Bengaluru, 560012, India, Department of Sciences and Humanities, CHRIST University, Mysore Road, Bangalore, 560 074, India

Rights

Restricted Access; Hardcopy may be available in the library

Relation

ISSN: 255874;

Format

online

Language

English

Type

Article

Identifier

https://doi.org/10.1007/s00209-025-03854-z

https://www.scopus.com/pages/publications/105016754081?origin=resultslist