Reflection groups in Lobachevsky spaces and the denominator identity for Lorentzian Kac-Moody algebras

Izvestiya Mathematics

Volumn 60, Issue 2, 1996, Pages 305-334

  Nikulin, V V ^a  

^a STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000526432     PISSN: 10645632     EISSN: 14684810     Source Type: Journal    
DOI: 10.1070/IM1996v060n02ABEH000072     Document Type: Article

References (39)

1. R. E. Borcherds.Lattices like the Leech lattice, J. Algebra 130 (1990), 219-234.
2. R. E. Borcherds.Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501-512.
3. R. E. Borcherds.The monster Lie algebra, Adv. Math. 83 (1990), 30-47.
4. R. E. Borcherds.The monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444.
5. s+2,2(ℝ) and infinite products, Invent. Math. 120 (1995), 161-213.
6. R. E. Borcherds.The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Preprint, Univ. of California, Berkeley 1994, pp. 1-12.
7. J. H. Conway.The automorphism group of the 26 dimensional even Lorentzian lattice, J. Algebra 80 (1983), 159-163.
8. F. Esselmann.Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe, Preprint no. 23, Univ. Bielefeld, 1992.
9. L. Greenberg.Finiteness theorems for Fuchsian and Kleinian groups, Discrete groups and automorphic functions (W. J. Harvey, ed.), Acad. Press, London 1977, pp. 199-257.
10. V. A. Gritsenko.Jacobi functions of n-variables, Zap. Nauch. Sem. LOMI 168 (1988), 32-45; English transl., J. Soviet Math. 53 (1991), 243-252.
11. V. A. Gritsenko.Modular forms and moduli spaces of abelian and K3 surfaces, Mathematica Gottingensis Schrift, des SFB 'Geometrie und Analysis' 26 (1993), 1-32; Russian transl., St. Petersburg Math. J. 6:6 (1994), 65-102.
12. V. A. Gritsenko and V. V. Nikulin.Siegel automorphic form correction of some Lorentzian Kac-Moody algebras, Mathematica Gottingensis Schrift, des SFB 'Geometrie und Analysis' 17 (1995), 1-26; Duke e-prints:alg-geom/9504006 (to appear in Amer. J. Math.).
13. V. G. Kac, Infinite dimensional Lie algebras, Cambridge Univ. Press 1990.
14. A. G. Khovanskii.Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevsky space, Functional Anal, i Prilozhen 20 (1986), 50-61; English transl. in Functional Anal. Appl. 20 (1986).
15. S. J. Kovács.The cone of curves of a K3 surface, Preprint, Univ. of Utah, 1993.
16. V. V. Nikulin.On factor groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections, Dokl. Akad. Nauk SSSR 248 (1979), 1307-1309; English transl. in Soviet Math. Dokl. 20 (1979), 1156-1158.
17. V. V. Nikulin.On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections, Algebraic-geometric applications, Current Problems in Math. Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii (1981), Moscow 3-114; English transl., J. Soviet Math. 22 (1983), 1401-1476.
18. V. V. Nikulin.Surfaces of type K3 with finite automorphism group and Picard group of rank three, Trudy Inst. Steklov 165 (1984), 113-142; English transl., Proc. Steklov. Inst. Math. 3 (1985), 131-157.
19. V. V. Nikulin.On arithmetic groups generated by reflections in Lobachevsky spaces, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 637-669; English transl., Math. USSR Izv. 16 (1981), 573-601.
20. V. V. Nikulin.On the classification of arithmetic groups generated by reflections in Lobachevsky spaces, Izv. Akad. Nauk SSSR Ser. Mat. 45:1 (1981), 113-142; English transl., Math. USSR Izv. 18 (1982), 99-123.
21. V. V. Nikulin.On a description of the automorphism groups of Enriques surfaces, Dokl. Akad. Nauk SSSR 277 (1984), 1324-1327; English transl., Soviet Math. Dokl. 30 (1984), 282-285.
22. V. V. Nikulin.Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proc. Int. Congr. Math., vol. 1, Berkeley 1986, pp. 654-669.
23. V. V. Nikulin.Algebraic surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces, Preprint no. 28, Max-Planck-Institut für Mathematik, Bonn 1989, pp. 1-126.
24. V. V. Nikulin.Basis of the diagram method for generalized reflection groups in Lobachevsky spaces and algebraic surfaces with nef anticanonical class, Int. J. Math. 7:1 (1996); Duke e-prints:alg-geom/9405011.
25. V. V. Nikulin.A lecture on Kac-Moody Lie algebras of the arithmetic type, Preprint no. 16, Queen's University, Canada 1994, pp. 1-12; Duke e-prints: alg-geom/9412003.
26. V. V. Nikulin.Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac-Moody Lie algebras, Mathematica Gottingensis Schrift, des SFB 'Geometrie und Analysis' 13 (1995), 1-30; Duke e-prints:alg-geom/9503003.
27. M. N. Prokhorov.The absence of discrete reflection groups with non-compact fundamental polyhedron of finite volume in Lobachevsky spaces of large dimension, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 413-424; English transl. in Math. USSR Izv. 28 (1987).
28. O. P. Ruzmanov.Reflection subgroups of Bianchi groups, Uspekhi Mat. Nauk 45:1 (1990), 189-190; English transl., Russian Math. Surveys 45 (1990), 227-228.
29. R. Scharlau and C. Walhorn.Integral lattices and hyperbolic reflection groups, Astérisque 209 (1992), 279-291.
30. E. B. Vinberg.Discrete groups generated by reflections in Lobachevsky spaces, Mat. Sb. (N.S.) 72 (1967), 471-488; English transl. in Math USSR 56. 1 (1967), 429-444.
31. É. B. Vinberg.On groups of unit elements of certain quadratic forms, Mat. Sbornik 87 (1972), 18-36; English transl., Math USSR Sbornik 16 (1972), 17-35.
32. E. B. Vinberg.The absence of crystallographic reflection groups in Lobachevsky spaces of large dimension, Trudy Moscow. Mat. Obshch. 47 (1984), 68-102; English transl. in Trans. Moscow Math. Soc. 47 (1985).
33. É. B. Vinberg.Hyperbolic reflection groups, Uspekhi Mat. Nauk 40 (1985), 29-66; English transl. in Russian Math. Surveys 40 (1985).
34. É. B. Vinberg.Discrete reflection groups in Lobachevsky spaces, Proc. Int. Congr. Math., Warsaw 1983, pp. 593-601.
35. É. B. Vinberg and O. V. Shvartsman.Discrete groups of motions of spaces of constant curvature, Sovrem. Prob. Mat. Fund. Napr. Vol. 29, Geometriya 2, VINITI, Moscow 1988, pp. 147-259; English transl. in Encyclopaedia of Math. Sciences. Geometry II, vol. 29, Springer-Verlag, Berlin 1991.
36. 1V. A. Gritsenko and V. V. Nikulin.K3 surfaces, Lorentzian Kac-Moody algebras and mirror symmetry, Mathematica Gottingensis Schrift, des SFB 'Geometrie und Analysis' 36 (1995), 1-15; Duke e-prints:alg-geom/9510008 (to appear in Math. Res. Letters).
37. V. V. Nikulin, Duke e-prints: alg-geom/9512018 1995, pp. 1-6.
38. V. A. Gritsenko and V. V. Nikulin, Duke e-prints: alg-geom/9603010 1996, pp. 1-40.
39. V. A. Gritsenko and V. V. Nikulin.Siegel automorphic form correction of a Lorentzian Kac-Moody algebra; Comp. Rend. Acad. Sci. Paris Ser. A-B, 321 (1995), 1151-1156.