Modified Nakajima monomials and the crystal $B(\infty )$

Abstract

In this paper, we give a new realization of the crystal $B(\infty )$ using modified Nakajima monomials. For the quantum affine algebras of type $A_n^{(1)}$, we give an explicit characterization of the monomials in $B(\infty )$, and construct a natural crystal isomorphism between monomial realization and path realization. As a corollary, we obtain a new realization of the crystal $B(\infty )$ for the quantum classical algebras of type $A_n$.