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

Abstract

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $\lambda=\Lambda_{1} - \Lambda_{2}$, where $\Lambda_{1}$, $\Lambda_{2}$ 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 $\lambda$.