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

Abstract

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