Crystal bases of q-deformed Kac modules over the quantum superalgebra $U_q(\text{gl}(m\mathrm{\mid }n))$

We introduce the notion of a crystal base of a finite dimensional q-deformed Kac module over the quantum superalgebra $U_q(\text{gl}(m\mathrm{\mid }n))$, and prove its existence and uniqueness. In particular, we obtain the crystal base of a finite dimensional irreducible $U_q(\text{gl}(m\mathrm{\mid }n))$-module with typical highest weight. We also show that the crystal base of a q-deformed Kac module is compatible with that of its irreducible quotient $V(\lambda )$ given by Benkart, Kang and Kashiwara when $V(\lambda )$ is an irreducible polynomial representation.