Crystal bases for the quantum queer superalgebra

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q(\mathfrak{q}(n))$. We define the notion of crystal bases and prove the tensor product rule for $U_q(\mathfrak{q}(n))$-modules in the category $O_{i}n{t}^{\ge 0}$. Our main theorem shows that every $U_q(\mathfrak q(n))$-moduleinthecategory$O_int^{\geq 0}$ has a unique crystal basis.