Polyhedral realization of the crystal bases for extremal weight modules over quantized hyperbolic Kac-Moody algebras of rank 2

TL;DR

This paper provides a geometric, polyhedral model for the crystal bases of extremal weight modules over rank 2 hyperbolic Kac-Moody algebras, expanding understanding of their structure in non-dominant cases.

Contribution

It introduces a novel polyhedral realization of crystal bases for extremal weight modules in hyperbolic Kac-Moody algebras of rank 2, covering non-dominant weight cases.

Findings

Polyhedral models for crystal bases are constructed.

The realization applies to extremal weights outside dominant/antidominant orbits.

Enhances geometric understanding of hyperbolic Kac-Moody algebra representations.

Abstract

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $\lambda$, where $\lambda$ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight.