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

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.

arXiv comment31 pages, 2 diagrams. arXiv admin note: text overlap with arXiv:1712.01009, arXiv:1702.02320 by other authors