Crystal Bases of q-deformed Kac Modules Over the Quantum Superalgebra Uq(𝔤(m|n))

We introduce the notion of a crystal base of a finite-dimensional q-deformed Kac module over the quantum superalgebra $U_q(\mathfrak {gl}(m|n))$, and prove its existence and uniqueness. In particular, we obtain the crystal base of a finite-dimensional irreducible $U_q(\mathfrak {gl}(m|n))$-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 (λ) given by Benkart, Kang and Kashiwara when V (λ) is an irreducible polynomial representation.