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 Tr. Mat. Inst. Steklova, 2000, Volume 230, Pages 3–255 (Mi tm509)

On the Classification of Hyperbolic Root Systems of Rank Three

V. V. Nikulin

Steklov Mathematical Institute, Russian Academy of Sciences

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2000, 230:3, 1–241

Bibliographic databases:
UDC: 512.7+512.81
Received in August 1999

Citation: V. V. Nikulin, “On the Classification of Hyperbolic Root Systems of Rank Three”, Tr. Mat. Inst. Steklova, 230, Nauka, MAIK «Nauka/Inteperiodika», M., 2000, 3–255; Proc. Steklov Inst. Math., 230:3 (2000), 1–241

Citation in format AMSBIB
\Bibitem{Nik00} \by V.~V.~Nikulin \paper On the Classification of Hyperbolic Root Systems of Rank Three \serial Tr. Mat. Inst. Steklova \yr 2000 \vol 230 \pages 3--255 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm509} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1802343} \zmath{https://zbmath.org/?q=an:1115.17302|0997.17014} \transl \jour Proc. Steklov Inst. Math. \yr 2000 \vol 230 \issue 3 \pages 1--241 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Gritsenko, V. V. Nikulin, “On classification of Lorentzian Kac–Moody algebras”, Russian Math. Surveys, 57:5 (2002), 921–979
2. Allcock D., “On the Y–555 complex reflection group”, Journal of Algebra, 322:5 (2009), 1454–1465
3. Ivashchuk V.D., Melnikov V.N., “On the Billiard Approach in Multidimensional Cosmological Models”, Gravitation & Cosmology, 15:1 (2009), 49–58
4. Vladimir D. Ivashchuk, Vitaly N. Melnikov, “On Brane Solutions Related to Non-Singular Kac–Moody Algebras”, SIGMA, 5 (2009), 070, 34 pp.
5. Chapovalov D., Chapovalov M., Lebedev A., Leites D., “The Classification of Almost Affine (Hyperbolic) Lie Superalgebras”, J Nonlinear Math Phys, 17, Suppl. 1 (2010), 103–161
6. Ivashchuk V.D., Melnikov V.N., “Black Brane Solutions Related to Non-Singular Kac-Moody Algebras”, Gravitation & Cosmology, 17:1 (2011), 7–17
7. [Anonymous], “The Reflective Lorentzian Lattices of Rank 3 Introduction”, Mem. Am. Math. Soc., 220:1033 (2012), VII+
8. Lakeland G.S., “Dirichlet-Ford Domains and Arithmetic Reflection Groups”, Pac. J. Math., 255:2 (2012), 417–437
9. Allcock D., “Root Systems For Lorentzian Kac-Moody Algebras in Rank 3”, Bull. London Math. Soc., 47:2 (2015), 325–342
10. Mark A., “Reflection Groups of the Quadratic Form -Px(0)(2) + X(1)(2) + ... X(N)(2) With P Prime”, Publ. Mat., 59:2 (2015), 353–372
11. Shimada I., “An Algorithm To Compute Automorphism Groups of K3 Surfaces and An Application To Singular K3 Surfaces”, Int. Math. Res. Notices, 2015, no. 22, 11961–12014
12. Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475
13. N. V. Bogachev, “Reflective anisotropic hyperbolic lattices of rank 4”, Russian Math. Surveys, 72:1 (2017), 179–181
14. Tomie M., Yoshii Y., “Reduced Hyperbolic Root Systems of Rank2”, J. Lie Theory, 27:2 (2017), 469–499
15. Ivashchuk V.D., “On Brane Solutions With Intersection Rules Related to Lie Algebras”, Symmetry-Basel, 9:8 (2017), 155
16. Gritsenko V. Nikulin V.V., “Lorentzian Kac-Moody Algebras With Weyl Groups of 2-Reflections”, Proc. London Math. Soc., 116:3 (2018), 485–533
17. Linowitz B., “Bounds For Arithmetic Hyperbolic Reflection Groups in Dimension 2”, Transform. Groups, 23:3 (2018), 743–753
18. N. V. Bogachev, A. Yu. Perepechko, “Vinberg's Algorithm for Hyperbolic Lattices”, Math. Notes, 103:5 (2018), 836–840
19. N. V. Bogachev, “Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4”, Izv. Math., 83:1 (2019), 1–19
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