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

It was recently understood that from the point of view of automorphic Lorentzian Kac-Moody algebras and some aspects of Mirror Symmetry, interesting hyperbolic root systems should have restricted arithmetic type and a generalized lattice Weyl vector. One can consider hyperbolic root systems with these properties as an appropriate hyperbolic analog of the classical finite and affine root systems. This series of papers is devoted to classification of hyperbolic root systems of restricted arithmetic type and with a generalized lattice Weyl vector $\rho$, having the rank 3 (it is the first non-trivial rank). In the Part I we announce classification of the maximal hyperbolic root systems of elliptic (i.e. $\rho^2 >0$) and parabolic (i.e. $\rho^2=0$) type, having the rank 3. We give sketch of the proof. Details of the proof and further results (non-maximal cases) will be given in Part II. Classification for hyperbolic type (i.e. $\rho^2<0$) and applications will be considered in Part III.