Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

TL;DR

This paper establishes a crystal isomorphism between Lakshmibai-Seshadri path crystals and extremal weight module crystals over rank 2 hyperbolic Kac-Moody algebras, providing computational tools for weight multiplicities.

Contribution

It proves the isomorphism of crystal components for extremal weight modules and LS path crystals, and offers an algorithm for weight multiplicity calculations in symmetric cases.

Findings

Connected components of crystal bases are isomorphic under certain conditions.

An explicit algorithm for counting elements of specific weights is developed.

The results apply to symmetric hyperbolic Kac-Moody algebras.

Abstract

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $\lambda$ be an arbitrary integral weight. We denote by $\mathbb{B}(\lambda)$ the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. Let $V(\lambda)$ be the extremal weight module of extremal weight $\lambda$ generated by the (cyclic) extremal weight vector $v_\lambda$ of weight $\lambda$, and let $\mathcal{B}(\lambda)$ be the crystal basis of $V(\lambda)$ with $u_\lambda \in \mathcal{B}(\lambda)$ the element corresponding to $v_\lambda$. We prove that the connected component $\mathcal{B}_0(\lambda)$ of $\mathcal{B}(\lambda)$ containing $u_\lambda$ is isomorphic, as a crystal, to the connected component $\mathbb{B}_0(\lambda)$ of $\mathbb{B}(\lambda)$ containing the straight line $\pi_\lambda$. Furthermore, we prove that if $\lambda$ satisfies a special condition, then the crystal basis $\mathcal{B}(\lambda)$ is isomorphic, as a crystal, to the crystal $\mathbb{B}(\lambda)$. As an application of these results, we obtain an algorithm for computing the number of elements of weight $\mu$ in $\mathcal{B}(\Lambda_1-\Lambda_2)$, where $\Lambda_1, \Lambda_2$ are the fundamental weights, in the case that $\mathfrak{g}$ is symmetric.