On the Classification of Hyperbolic Root Systems of Rank Three

V. V. Nikulin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, On the classification of hyperbolic root systems of rank three, Tome 230 (2000), pp. 3-255

Résumé

This is the first monograph devoted to the classification of hyperbolic root systems which are important for the theory of Lorentzian (or hyperbolic) Kac–Moody algebras. These hyperbolic root systems should have a restricted arithmetic type and a generalized lattice Weyl vector. One can consider them as an appropriate hyperbolic analogue of finite and affine root systems. The author obtained the finiteness results for the hyperbolic root systems. The classification of these root systems is considered for the first nontrivial and the most rich case of rank three. It requires very nontrivial and long calculations. One can consider this work as the starting point for developing the complete theory of Lorentzian Kac–Moody algebras for the rank three case. The rank three case is the hyperbolic analogue of $sl_2$. For scientists, senior students, and postgraduates interested in the theory of Lie groups and algebras, algebraic geometry, and mathematical and theoretical physics.