On the classification of hyperbolic root systems of the rank three. Part II

Details On the classification of hyperbolic root systems of the rank three. Part II On the classification of hyperbolic root systems of the rank three. Part II

1997-12-30

arxiv.org/abs/alg-geom/9712033v2

Physics

High Energy Physics

High Energy Physics - Theory

AMS-Tex, 66 pages, no figures. Significantly improved variant: estimates for the method of narrow places of polyhedra are very

Scientific paper

Here we prove classification results announced in Part I (alg-geom/9711032). We classify maximal hyperbolic root systems of the rank 3 having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2\ge 0$ (i.e. of elliptic or parabolic type). We give classification of all reflective of elliptic or parabolic type elementary hyperbolic lattices of the rank three. We apply the same method (narrow places of polyhedra) which was developed to prove finiteness results on reflective hyperbolic lattices. We also use some additional arithmetic arguments: studying of class numbers of central symmetries. The same methods permit to get similar results for hyperbolic type. We will consider hyperbolic type in Part III. These results are important for Theory of Lorentzian Kac--Moody algebras and some aspects of Mirror Symmetry.

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