ABSTRACT. We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Saçlioglu, we present the 238 hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.
Let g be a Kac-Moody Lie algebra. We give an interpretation of Tits' associated group functor using representation theory of g and we construct a locally compact "Kac-Moody group" G over a finite field k. Using (twin) BN -pairs (G, B, N ) and (G, B − , N) for G we show that if k is "sufficiently large", then the subgroup B − is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and nonuniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.
We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W pnq and Spnq of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W pgq and Spgq, where g denotes the Kac-Moody algebra Ar, Dr or Er. Then W pAn´1q and SpAn´1q are the Lie superalgebras W pnq and Spnq. The algebras W pgq and Spgq are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras Bpgq obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W pArq and SpArq in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level ě´2. The analogous definitions of the algebras in the D-and E-series are given. In the latter case the full set of defining relations is conjectured.
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