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

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