Generalized $q$-boson algebras and their integrable modules

TL;DR

This paper introduces a generalized $q$-boson algebra linked to Nichols algebras and explores its integrable modules, extending previous work on $q$-boson algebra modules.

Contribution

It defines a new class of generalized $q$-boson algebras and studies their integrable modules, broadening the understanding of their structure and representation theory.

Findings

Defined the generalized $q$-boson algebra $ ext{B}$ associated with Nichols algebras.

Extended results on integrable modules from the Kashiwara and Nakashima framework.

Provided new insights into the structure and representation theory of these algebras.

Abstract

We define the generalized $q$-boson algebra $ \cmdB$ associated to a pair of Nichols algebras and a skew pairing. We study integrable $ \cmdB$-modules, generalizing results by M. Kashiwara and T. Nakashima on integrable modules over a $q$-boson (Kashiwara) algebra.