Path model for a level-zero extremal weight module over a quantum affine algebra II

Let ϖi be a level-zero fundamental weight for an affine Lie algebra g over Q, and let B(ϖi) be the crystal of all Lakshmibai–Seshadri paths of shape ϖi. First, we prove that the crystal graph of B(ϖi) is connected. By combining this fact with the main result of our previous work, we see that B(ϖi) is, as a crystal, isomorphic to the crystal base B(ϖi) of the extremal weight module V(ϖi) over a quantum affine algebra Uq(g) over Q(q) of extremal weight ϖi. Next, we obtain an explicit description of the decomposition of the crystal B(mϖi) of all Lakshmibai–Seshadri paths of shape mϖi into connected components. Furthermore, we prove that B(mϖi) is, as a crystal, isomorphic to the crystal base B(mϖi) of the extremal weight module V(mϖi) over Uq(g) of extremal weight mϖi.