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 2000, Volume 230

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On the classification of hyperbolic root systems of rank three

Author: V. V. Nikulin
Volume Editor: I. R. Shafarevich
Editor in Chief: E. F. Mishchenko

ISBN: 5-02-002502-X, 5-7846-0084-2

Abstract: 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.

Citation: V. V. Nikulin, On the classification of hyperbolic root systems of rank three, Tr. Mat. Inst. Steklova, 230, ed. I. R. Shafarevich, E. F. Mishchenko, Nauka, MAIK «Nauka/Inteperiodika», M., 2000, 256 pp.

Citation in format AMSBIB:
\Bibitem{1} \by V.~V.~Nikulin \book On the classification of hyperbolic root systems of rank three \serial Tr. Mat. Inst. Steklova \yr 2000 \vol 230 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \ed I.~R.~Shafarevich, E.~F.~Mishchenko \totalpages 256 \mathnet{http://mi.mathnet.ru/book243} 

 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.