Imaginary Verma Modules for $U_q(\widehat{\mathfrak{sl}(2)})$ and Crystal-like bases

TL;DR

This paper introduces the concept of imaginary crystal bases for imaginary Verma modules of the quantum affine algebra U_q(sl(2)), establishing their existence within a specific module category.

Contribution

It defines and proves the existence of imaginary crystal bases for reduced imaginary Verma modules of U_q(sl(2)), extending crystal basis theory to imaginary modules.

Findings

Existence of imaginary crystal bases for certain modules.

Construction of the imaginary crystal basis using Kashiwara algebra K_q.

Extension of crystal basis concepts to imaginary Verma modules.

Abstract

We consider imaginary Verma modules for quantum affine algebraU_q(\widehat{\mathfrak{sl}(2)}) and define a crystal-like base which we call an imaginary crystal basis using the Kashiwara algebra K_q constructed in earlier work of the authors. In particular, we prove the existence of imaginary like bases for a suitable category of reduced imaginary Verma modules for U_q(\widehat{\mathfrak{sl}(2)}).