Modified Nakajima monomials and the crystal B(∞)

In this paper, we give a new realization of the crystal B(∞) using modified Nakajima monomials. For the quantum affine algebras of type , we give an explicit characterization of the monomials in B(∞), and construct a natural crystal isomorphism between monomial realization and path realization. As a corollary, we obtain a new realization of the crystal B(∞) for the quantum classical algebras of type An.