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

TL;DR

This paper studies the structure of extremal weight modules over a rank 2 hyperbolic Kac-Moody algebra, establishing their crystal basis properties, irreducibility, and connection to Lakshmibai-Seshadri paths.

Contribution

It provides a detailed analysis of the crystal basis and structure of extremal weight modules over hyperbolic Kac-Moody algebras of rank 2, including their connectedness and isomorphism to path models.

Findings

The crystal basis is connected.

Each weight space in the crystal basis is finite.

The module is irreducible.

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 $\lambda$.