Category O for quantum loop algebras

TL;DR

This paper extends the category O framework to quantum loop algebras associated with arbitrary Kac-Moody algebras, providing explicit module realizations and new computational tools for q-characters.

Contribution

It introduces a generalized category O for quantum loop algebras, including explicit simple module realizations and novel methods for q-character computation.

Findings

Generalization of category O to quantum loop algebras for arbitrary Kac-Moody g

Explicit realizations of all simple modules in the new setting

Development of new tools for q-character computation, applicable even in finite type cases

Abstract

We generalize the Hernandez-Jimbo category O of representations of Borel subalgebras of quantum affine algebras to the case of quantum loop algebras for arbitrary Kac-Moody g (as well as related algebras, such as quantum toroidal gl_1). Moreover, we give explicit realizations of all simple modules, and devise tools for the computation of q-characters that are new even for g of finite type. Our techniques allow us to generalize classic results of Frenkel-Hernandez, Frenkel-Mukhin, Hernandez-Jimbo and Hernandez-Leclerc, as well as prove conjectures of Feigin-Jimbo-Miwa-Mukhin and Mukhin-Young.