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 $\lambda ={\Lambda }_1-{\Lambda }_2,$ where ${\Lambda }_1,{\Lambda }_2$ are the fundamental weights. Denote by $V(\lambda )$ the extremal weight module of extremal weight λ 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 μ in $\mathcal{B}(\lambda )$ is a finite set for every integral weight μ, and $\mathcal{B}{(\lambda )}_{\lambda }=\left\{u_{\lambda }\right\},$ (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 λ.

Keywords:

• Crystal basis
• extremal weight module
• Kac-Moody algebra
• Lakshmibai-Seshadri path
• path model
• quantum group

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B37
• 17B67
• 81R50