Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $\lambda ={\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_2$, where ${\mathrm{\Lambda }}_1$, ${\mathrm{\Lambda }}_2$ are the fundamental weights. Denote by $V(\lambda )$ the extremal weight module of extremal weight $\lambda$ with $v_{\lambda }$ the extremal weight vector, and by $\mathcal{B}(\lambda )$ the crystal basis of $V(\lambda )$ with $u_{\lambda }$ the element corresponding to $v_{\lambda }$. We prove that (i) $\mathcal{B}(\lambda )$ is connected, (ii) the subset $\mathcal{B}(\lambda {)}_{\mu }$ of elements of weight $\mu$ in $\mathcal{B}(\lambda )$ is a finite set for every integral weight $\mu$, and $\mathcal{B}(\lambda {)}_{\lambda }=\{u_{\lambda }\}$, (iii) every extremal element in $\mathcal{B}(\lambda )$ is contained in the Weyl group orbit of $u_{\lambda }$, (iv) $V(\lambda )$ is irreducible. Finally, we prove that the crystal basis $\mathcal{B}(\lambda )$ is isomorphic, as a crystal, to the crystal $\mathbb{B}(\lambda )$ of Lakshmibai-Seshadri paths of shape $\lambda$.