Imaginary Verma modules for Uq(sl(2)ˆ) and crystal-like bases

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