Imaginary Verma modules and Kashiwara algebras for $U_q(\widehat{\mathfrak{sl}(2)})$

TL;DR

This paper explores the structure of imaginary Verma modules for the quantum affine algebra $U_q(\\hat{\rak{sl}(2)})$, introducing Kashiwara operators and demonstrating the simplicity of a related quotient module.

Contribution

It constructs Kashiwara type operators and the Kashiwara algebra for the quantum affine algebra, and proves the simplicity of a specific quotient module.

Findings

Construction of Kashiwara operators and algebra for $U_q(\\hat{\rak{sl}(2)})$

Identification of a simple quotient module $\\mathcal N_q^-$

Advancement in understanding modules over quantum affine algebras

Abstract

We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{sl}(2)})$ and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{sl}(2)})$ is a simple $\mathcal K_q$-module.