Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Abstract

Let be a hyperbolic Kac-Moody algebra of rank 2, and set where are the fundamental weights. Denote by the extremal weight module of extremal weight λ with the extremal weight vector, and by the crystal basis of with the element corresponding to We prove that (i) is connected, (ii) the subset of elements of weight μ in is a finite set for every integral weight μ, and (iii) every extremal element in is contained in the Weyl group orbit of (iv) is irreducible. Finally, we prove that the crystal basis is isomorphic, as a crystal, to the crystal of Lakshmibai-Seshadri paths of shape λ.