On the Classification of Hyperbolic Root Systems of Rank Three

V. V. Nikulin

Informatics and Automation, 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.

Bibliographie

[1] Alekseev V.A., Nikulin V.V., “Klassifikatsiya poverkhnostei del Petstso s log-terminalnymi osobennostyami indeksa $\le 2$, involyutsii na poverkhnostyakh K3 i gruppy otrazhenii v prostranstvakh Lobachevskogo”, Dokl. po matematike i prilozheniyam MIAN, 2 (1988), 51–150 | MR

[2] Alekseev V.A., Nikulin V.V., “Klassifikatsiya poverkhnostei del Petstso s log-terminalnymi osobennostyami indeksa $\le 2$ i involyutsii na poverkhnostyakh K3”, DAN SSSR, 306:3 (1989), 525–528 | MR | Zbl

[3] Borcherds R., “Generalized Kac–Moody algebras”, J. Algebra, 115 (1988), 501–512 | DOI | MR | Zbl

[4] Borcherds R., “The monster Lie algebra”, Adv. Math., 83 (1990), 30–47 | DOI | MR | Zbl

[5] Borcherds R., “The monstrous moonshine and monstrous Lie superalgebras”, Invent. math., 109 (1992), 405–444 | DOI | MR | Zbl

[6] Borcherds R., “Sporadic groups and string theory”, Proc. Eur. Congr. Math. 1992, 411–421 | MR | Zbl

[7] Borcherds R., “Automorphic forms on $O_{s+2,2}$ and infinite products”, Invent. math., 120 (1995), 161–213 | DOI | MR | Zbl

[8] Borcherds R., “The moduli space of Enriques surfaces and the fake monster Lie superalgebra”, Topology, 35:3 (1996), 699–710 | DOI | MR | Zbl

[9] Borevich Z.I., Shafarevich I.R., Teoriya chisel, Nauka, M., 1985 | MR | Zbl

[10] Cassels J.W.S., Rational quadratic forms, Acad. Press, London, 1978 | MR | Zbl

[11] Eichler M., Quadratische Formen und orthogonale Gruppen, Springer-Verl., Berlin, 1952 | MR

[12] Esselmann F., “Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe”, J. Number Theory, 61:1 (1996), 103–144 | DOI | MR | Zbl

[13] Gritsenko V.A., Nikulin V.V., “Siegel automorphic form correction of some Lorentzian Kac–Moody Lie algebras”, Amer. J. Math., 119:1 (1997), 181–224 ; alg-geom/9504006 | DOI | MR | Zbl

[14] Gritsenko V.A., Nikulin V.V., “Siegel automorphic form correction of a Lorentzian Kac–Moody algebra”, C. r. Acad. sci. Paris. A–B, 321 (1995), 1151–1156 | MR | Zbl

[15] Gritsenko V.A., Nikulin V.V., “K3 surfaces, Lorentzian Kac–Moody algebras and mirror symmetry”, Math. Res. Lett., 3:2 (1996), 211–229 ; alg-geom/9510008 | MR | Zbl

[16] Gritsenko V.A., Nikulin V.V., “Modulyarnye formy Iguzy i “prosteishie” lorentsevy algebry Katsa–Mudi”, Mat. sb., 187:11 (1996), 27–66 ; alg-geom/9603010 | MR | Zbl

[17] Gritsenko V.A., Nikulin V.V., “Automorphic forms and Lorentzian Kac–Moody algebras. Part I”, Intern. J. Math., 8:2 (1998), 153–199 ; alg-geom/9610022 | DOI | MR

[18] Gritsenko V.A., Nikulin V.V., “Automorphic forms and Lorentzian Kac–Moody algebras. Part II”, Intern. J. Math., 8:2 (1998), 201–275 ; alg-geom/9611028 | DOI | MR

[19] Gritsenko V.A., Nikulin V.V., “The arithmetic mirror symmetry and Calabi–Yau manifolds”, Commun. Math. Phys., 210 (2000), 1–11 ; alg-geom/9612002 | DOI | MR | Zbl

[20] Harvey J., Moore G., “Algebras, BPS-states, and strings”, Nucl. Phys. B, 463 (1996), 315–368 ; hep-th/9510182 | DOI | MR | Zbl

[21] Harvey J., Moore G., On the algebras of BPS-states, Preprint , 1996 hep-th/9609017 | MR

[22] Kac V., Infinite dimensional Lie algebras, Cambridge Univ. Press, Cambridge, 1990 | MR

[23] Kawai T., “String duality and modular forms”, Phys. Lett. B, 397 (1997), 51–62 ; hep-th/9607078 | DOI | MR

[24] Kawai T., K3 surfaces, Igusa cusp forms and string theory, , 1997 hep-th/9710016 | MR

[25] Kneser M., “Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen”, Arch. Math. Basel, 7 (1956), 323–332 | DOI | MR | Zbl

[26] Moore G., String duality, automorphic forms, and generalized Kac–Moody algebras, , 1997 hep-th/9710198 | MR

[27] Nikulin V.V., “Tselochislennye simmetricheskie bilineinye formy i nekotorye ikh geometricheskie prilozheniya”, Izv. AN SSSR. Ser. mat., 43:1 (1979), 111–177 | MR | Zbl

[28] Nikulin V.V., “O faktor-gruppakh grupp avtomorfizmov giperbolicheskikh form po podgruppam, porozhdennym 2-otrazheniyami”, DAN SSSR, 248 (1979), 1307–1309 | MR | Zbl

[29] Nikulin V.V., “O faktor-gruppakh grupp avtomorfizmov giperbolicheskikh form po podgruppam, porozhdennym 2-otrazheniyami. Algebro-geometricheskie prilozheniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 18, VINITI, M., 1981, 3–114 | MR

[30] Nikulin V.V., “Ob arifmeticheskikh gruppakh, porozhdennykh otrazheniyami, v prostranstvakh Lobachevskogo”, Izv. AN SSSR. Ser. mat., 44:3 (1980), 637–668 | MR

[31] Nikulin V.V., “O klassifikatsii arifmeticheskikh grupp, porozhdennykh otrazheniyami, v prostranstvakh Lobachevskogo”, Izv. AN SSSR. Ser. mat., 45:1 (1981), 113–142 | MR | Zbl

[32] Nikulin V.V., “Poverkhnosti tipa K3 s konechnoi gruppoi avtomorfizmov i gruppoi Pikara ranga tri”, Tr. MIAN, 165, 1984, 119–142 | MR | Zbl

[33] Nikulin V.V., “Ob opisanii grupp avtomorfizmov poverkhnostei Enrikvesa”, DAN SSSR, 277 (1984), 1324–1327 | MR | Zbl

[34] Nikulin V.V., “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces”, Proc. Intern. Congr. Math. (Berkeley, 1986), vol. 1, 654–669 | MR

[35] Nikulin V.V., “Basis of the diagram method for generalized reflection groups in Lobachevsky spaces and algebraic surfaces with nef anticanonical class”, Intern. J. Math., 7:1 (1996), 71–108 ; alg-geom/9405011 | DOI | MR | Zbl

[36] Nikulin V.V., A lecture on Kac–Moody Lie algebras of the arithmetic type, Preprint Queen's Univ. (Canada) #1994-16, 1994; alg-geom/9412003

[37] Nikulin V.V., “Gruppy otrazhenii v prostranstvakh Lobachevskogo i tozhdestvo dlya znamenatelya lorentsevykh algebr Katsa–Mudi”, Izv. RAN. Ser. mat., 60:2 (1996), 73–106 ; alg-geom/9503003 | MR | Zbl

[38] Nikulin V.V., “The remark on discriminants of K3 surfaces moduli as sets of zeros of automorphic forms”, J. Math. Sci., 81:3 (1996), 2738–2743 ; alg-geom/9512018 | DOI | MR | Zbl

[39] Nikulin V.V., “K3 surfaces with interesting groups of automorphisms”, J. Math. Sci. (New York), 95:1 (1999), 2028–2048 ; alg-geom/9701011 | DOI | MR | Zbl

[40] Nikulin V.V., “A remark on algebraic surfaces with polyhedral Mori cone”, Nagoya J. Math., 157 (2000), 73–92 ; math.AG/9806047 | MR | Zbl

[41] Pyatetskii-Shapiro I.I., Shafarevich I.R., “Teorema Torelli dlya algebraicheskikh poverkhnostei tipa K3”, Izv. AN SSSR. Ser. mat., 35 (1971), 530–572 | Zbl

[42] Raghunatan M.S., Discrete subgroups of Lie groups, Springer-Verl., Berlin, 1972 | MR

[43] Ruzmanov O., On maximal arithmetic reflection groups of the hyperbolic plane, Preprint Bielefeld Univ. 91-043, 1991, 12 pp. | Zbl

[44] Scharlau R., Walhorn C., “Integral lattices and hyperbolic reflection groups”, Astérisque, 209 (1992), 279–291 | MR | Zbl

[45] Serre J.-P., Cours d'arithmétique, Press. Univ. France, Paris, 1970 | MR | Zbl

[46] Shanks D., “Class number, a theory of factorization, and genera”, Proc. Sympos. Pure Math., 20 (1970), 415–440 | MR

[47] Shvartsman O.V., “Ob odnom neobkhodimom uslovii reflektivnosti ternarnoi reshetki”, Problemy teorii grupp i gomologicheskoi algebry, Izd. Yarosl. gos. un-ta, Yaroslavl, 1988, 66–70 | MR

[48] Vinberg E.B., “Diskretnye gruppy, porozhdennye otrazheniyami v prostranstvakh Lobachevskogo”, Mat. sb., 72 (1967), 471–488 | MR | Zbl

[49] Vinberg E.B., “O gruppakh edinits nekotorykh kvadratichnykh form”, Mat. sb., 87 (1972), 18–36 | MR | Zbl

[50] Vinberg E.B., “Otsutstvie kristallograficheskikh grupp otrazhenii v prostranstvakh Lobachevskogo bolshoi razmernosti”, Tr. Mosk. mat. o-va, 47 (1984), 67–102 | MR

[51] Vinberg E.B., “Giperbolicheskie gruppy otrazhenii”, UMN, 40 (1985), 29–66 | MR | Zbl

[52] Vinberg E.B., Shvartsman O.V., “Diskretnye gruppy dvizhenii prostranstv postoyannoi krivizny”, Geometriya - 2, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 29, VINITI, M., 1988, 147–259 | MR

[53] Walhorn C., Arithmetische Spiegelungsgruppen auf dem 4-dimensionalen hyperbolischen Raum, Diss. zur Erlangung des Doktorgrades der Fakultät für Mathematik der Universität Bielefeld, 1993 | Zbl

[54] Watson G.L., “Transformations of a quadratic form which do not increase the class-number”, Proc. London Math. Soc., 12:48 (1962), 577–587 | DOI | MR | Zbl

[55] Nikulin V.V., “Teoriya lorentsevykh algebr Katsa–Mudi”, T. 8. Algebra, Trudy Mezhdunar. konf., posvyasch. 90-letiyu so dnya rozhdeniya L.S. Pontryagina, Itogi nauki i tekhniki. Sovremennaya matematika i ee prilozheniya, 69, VINITI, M., 1999, 149–167 ; math.AG/9810001