Crystal base of the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$

Abstract

We construct a crystal base of $U_q(\mathfrak{gl}(m|n))^-$, the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$. We give a combinatorial description of the associated crystal $\mathscr{B}_{m|n}(\infty)$, which is equal to the limit of the crystals of the ($q$-deformed) Kac modules $K(λ)$. We also construct a crystal base of a parabolic Verma module $X(λ)$ associated with the subalgebra $U_q(\mathfrak{gl}_{0|n})$, and show that it is compatible with the crystal base of $U_q(\mathfrak{gl}(m|n))^-$ and the Kac module $K(λ)$ under the canonical embedding and projection of $X(λ)$ to $U_q(\mathfrak{gl}(m|n))^-$ and $K(λ)$, respectively.