# Imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$-modules in category $\mathcal O^q_{\text{red,im}}$

## TL;DR

This paper establishes the existence of imaginary crystal bases for all modules in a specific category of quantum affine algebra representations, extending previous results that were limited to certain modules.

## Contribution

It generalizes the existence of imaginary crystal bases from reduced imaginary Verma modules to all modules in the category $ ext{O}^q_{ ext{red,im}}$ for $U_q( ext{sl}(2))$.

## Key findings

- Existence of imaginary crystal bases for all modules in the category.
- Extension of previous results from specific modules to the entire category.
- Provides a foundation for further study of module structures in quantum affine algebras.

## Abstract

Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for $U_q(\widehat{\mathfrak{sl}(2)})$. In this paper we show the existence of imaginary crystal basis for any object in the category $\mathcal O^q_{\text{red,im}}$.