Lakshmibai-Seshadri paths for hyperbolic Kac-Moody algebras of rank $2$

Abstract

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set $λ: = Λ_1 - Λ_2$, where $Λ_1, Λ_2$ are the fundamental weights for $\mathfrak{g}$; note that $λ$ is neither dominant nor antidominant. Let $\mathbb{B}(λ)$ be the crystal of all Lakshmibai-Seshadri paths of shape $λ$. We prove that (the crystal graph of) $\mathbb{B}(λ)$ is connected. Furthermore, we give an explicit description of Lakshmibai-Seshadri paths of shape $λ$.

15 pages