Abstract. We construct quantum deformations of enveloping algebras of Borcherds superalgebras, their Verma modules, and their irreducible highest weight modules.

Affine Kac-Moody algebras represent a well-trodden and well-understood littoral beyond which stretches the vast, chaotic, and poorly-understood ocean of indefinite Kac-Moody algebras. The simplest indefinite Kac-Moody algebras are the rank 2 Kac-Moody algebras (a) (a ≥ 3) with symmetric Cartan matrix , which form part of the class known as hyperbolic Kac-Moody algebras. In this paper, we probe deeply into the structure of those algebras (a), the e. coli of indefinite Kac-Moody algebras. Using Berman-Moody’s formula ([BM]), we derive a purely combinatorial closed form formula for the root multiplicities of the algebra (a), and illustrate some of the rich relationships that exist among root multiplicities, both within a single algebra and between different algebras in the class. We also give an explicit description of the root system of the algebra (a). As a by-product, we obtain a simple algorithm to find the integral points on certain hyperbolas.

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We construct quantum deformations of imaginary Verma modules over Ufalse(A1false(1false)false) and show that, for generic q, imaginary Verma modules over Ufalse(A1false(1false)false) can be deformed to those over the quantum group Uqfalse(A1false(1false)false) in such a way that the dimensions of the weight spaces are invariant under the deformation. We also prove the PBW theorem for Uqfalse(A1false(1false)false) with respect to the triangular decomposition induced from the root partition corresponding to the imaginary Verma modules. 1991 Mathematics Subject Classification: 17B67, 17B65, 17B10.

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