Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras

TL;DR

This paper introduces a new type of crystal base, called imaginary crystal base, for reduced imaginary Verma modules of untwisted quantum affine algebras, expanding the understanding of their structure.

Contribution

It defines the imaginary crystal base using Kashiwara algebra and proves its existence for modules in a specific category, advancing the theory of quantum affine algebra representations.

Findings

Existence of imaginary crystal base for reduced imaginary Verma modules

Construction of the base using Kashiwara algebra $\\mathcal K_q$

Extension of crystal base theory to new module categories

Abstract

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\mc{O}^q_{red,im}$ containing the reduced imaginary Verma modules for $U_q(\hat{\g})$.