Crystal Bases for the Quantum Superalgebra $U_q(D(N,1))$, $U_q(B(N,1))$

Abstract

Let $V(\lambda)$ be the irreducible lowest weight $U_q(D(N,1))$-module with lowest weight $\lambda$. Assume $\lambda = n_0\omega_0-\sum_{i=0}^{N}n_i\omega_i$, where $\omega_0$ is the fundamental weight corresponding to the unique odd coroot $h_0$, and $n_i$ are positive integers. $V(\lambda)$ is called typical if $n_0 \geq 0$. In this paper, we construct polarizable crystal bases of $V(\lambda)$ in the category ${\cal O}_{int}$, which is a class of integrable modules. We also describe the decomposition of the tensor product of typical representations into irreducible ones, using the generalized Littlewood-Richardson rule for $U_q(D(N))$. We also present analogous results for the quantum superalgebra $U_q(B(N,1))$.