Crystal bases of q-deformed Kac modules over the quantum superalgebra $U_q(\gl(m|n))$

Abstract

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