  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 $U_q(\hat{\g})$.

Representation Theory;Quantum Algebra