Crystal bases for the quantum queer superalgebra

Abstract

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 ${\mathcal{O}}_{\mathrm{int}}^{\ge 0}$. Our main theorem shows that every $U_q(\mathfrak{q}(n))$-module in the category ${\mathcal{O}}_{\mathrm{int}}^{\ge 0}$ has a unique crystal basis.