On the classification of hyperbolic root systems of rank three. (Q2701928)

This is the first monograph devoted to the classification of hyperbolic root systems, which are important for the theory of Lorentzian Kac-Moody algebras. These hyperbolic root systems should have a restricted arithmetic type and a generalized lattice Weyl vector. They can be considered as an appropriate hyperbolic analogue of finite and affine root systems. NEWLINENEWLINENEWLINEThere are three types of root systems: elliptic (there is a generalized lattice Weyl vector with positive square), parabolic (there is a generalized lattice Weyl vector with zero square and there is no generalized lattice Weyl vector with positive square) and hyperbolic. The number of maximal reflective root systems is finite for any rank \(\geq 3\). The author determines a list of all reflective elementary hyperbolic lattices of rank three. There are \(122\) principal and \(38\) nonprincipal reflective root systems of elliptic and parabolic types, and there are \(66\) principal and \(21\) nonprincipal reflective root systems of hyperbolic type. NEWLINENEWLINENEWLINEThe main method used is the method of narrow parts of convex polygons on a hyperbolic space [see the author, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 637-669 (1980; Zbl 0441.22008)]. The classification requires very nontrivial computer calculations. Computer programs are included in the monograph. The results are presented in seven tables.NEWLINENEWLINENEWLINEThe English translation is published in Proc. Steklov Inst. Math. 230, 241 p. (2000).