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Let 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 B (λ) the crystal basis of V (λ) with u λ the element corresponding to v λ. We prove that (i) B (λ) is connected, (ii) the subset B (λ) μ of elements of weight μ in B (λ) is a finite set for every integral weight μ, and B (λ) λ = { u λ } , (iii) every extremal element in B (λ) is contained in the Weyl group orbit of u λ , (iv) V (λ) is irreducible. Finally, we prove that the crystal basis B (λ) is isomorphic, as a crystal, to the crystal B (λ) of Lakshmibai-Seshadri paths of shape λ. [ABSTRACT FROM AUTHOR]

Let 픤 be a hyperbolic Kac-Moody algebra of rank 2, and set , where are the fundamental weights for 픤; note that λ is neither dominant nor antidominant. Let 픹(λ) be the crystal of all Lakshmibai-Seshadri paths of shape λ. We prove that (the crystal graph of) 픹(λ) is connected. Furthermore, we give an explicit description of Lakshmibai-Seshadri paths of shape λ. [ABSTRACT FROM AUTHOR]