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 $λ=Λ_{1} - Λ_{2}$, where $Λ_{1}$, $Λ_{2}$ are the fundamental weights. Denote by $V(λ)$ the extremal weight module of extremal weight $λ$ with $v_λ$ the extremal weight vector, and by $\mathcal{B}(λ)$ the crystal basis of $V(λ)$ with $u_λ$ the element corresponding to $v_λ$. We prove that (i) $\mathcal{B}(λ)$ is connected, (ii) the subset $\mathcal{B}(λ)_μ$ of elements of weight $μ$ in $\mathcal{B}(λ)$ is a finite set for every integral weight $μ$, and $\mathcal{B}(λ)_λ = \{u_λ\}$, (iii) every extremal element in $\mathcal{B}(λ)$ is contained in the Weyl group orbit of $u_λ$, (iv) $V(λ)$ is irreducible. Finally, we prove that the crystal basis $\mathcal{B}(λ)$ is isomorphic, as a crystal, to the crystal $\mathbb{B}(λ)$ of Lakshmibai-Seshadri paths of shape $λ$.

20 pages, 2 diagrams