Polyhedral Realization of Crystal Bases for Generalized Kac-moody Algebras

In this paper, we give polyhedral realization of the crystal B(∞) of U− q (g) for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of rank 2, 3 and Monster Lie algebras. Introduction In his study of Conway and Norton’s Moonshine Conjecture [3] for the infinite dimensional Z-graded representation V ♮ of the Monster sporadic simple group, Borcherds introduced a new class of infinite dimensional Lie algebras called the generalized Kac-Moody algebras [1, 2]. The structure and representation theories of generalized Kac-Moody algebras are very similar to those of Kac-Moody algebras, and a lot of facts about Kac-Moody algebras can be extended to generalized Kac-Moody algebras. The main difference is that the generalized Kac-Moody algebras may have simple roots with non-positive norms whose multiplicity can be greater than one, called imaginary simple roots, and they may have infinitely many simple roots. The quantum groups Uq(g) introduced by Drinfel’d and Jimbo, independently are q-deformations of the universal enveloping algebras U(g) of KacMoody algebras g [4, 7]. The important feature of quantum groups is that the representation theory of Uq(g) is the same as that of U(g). Therefore, to understand the structure of representations over Uq(g), it is enough to understand that of representations over Uq(g) for some special parameter q which is easy to treat. The crystal basis theory which can be viewed as the representation theory at q = 0 was introduced by Kashiwara [14]. Among others, he 2000 Mathematics Subject Classification. Primary 81R50, Secondary 17B37.