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

Abstract

Let be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by the crystal of all Lakshmibai-Seshadri paths of shape λ. Let be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector of weight λ, and let be the crystal basis of with the element corresponding to We prove that the connected component of containing is isomorphic, as a crystal, to the connected component of containing the straight line Furthermore, we prove that if λ satisfies a special condition, then the crystal basis is isomorphic, as a crystal, to the crystal As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in where are the fundamental weights, in the case that is symmetric.