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Journal of Mathematical Sciences

, Volume 106, Issue 4, pp 3212–3221 | Cite as

A Theory of Lorentzian Kac--Moody Algebras

  • V. V. Nikulin
Article

Keywords

Moody Algebra 
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REFERENCES

  1. 1.
    W. L. Baily, “Fourier-Jacobi series, ” In: Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math., Vol. IX (A. Borel andG. D. Mostow, eds.), Amer. Math. Soc., Providence, Rhode Island (1966), pp. 296-300.Google Scholar
  2. 2.
    R. Borcherds, “Generalized Kac-Moody algebras, ” J. Algebra, 115, 501-512 (1988).Google Scholar
  3. 3.
    R. Borcherds, “The monster Lie algebra, ” Adv. Math., 83, 30-47 (1990).Google Scholar
  4. 4.
    R. Borcherds, “The monstrous moonshine and monstrous Lie superalgebras, ” Invent. Math., 109, 405-444 (1992).Google Scholar
  5. 5.
    R. Borcherds, “Sporadic groups and string theory, ” In: Proc. European Congress Math. (1992), pp. 411-421.Google Scholar
  6. 6.
    R. Borcherds, “Automorphic forms on Os+2,2(R) and in.nite products, ” Invent. Math., 120, 161-213 (1995).Google Scholar
  7. 7.
    G. L. Cardoso, “Perturbative gravitational couplings and Siegel modular forms in D = 4, N = 2 string models, ” Nucl. Phys. Proc. Suppl., 56B, 94-101 (1997); hep-th/9612200.Google Scholar
  8. 8.
    G. L. Cardoso,G. Curio, andD. Lust, “Perturbative coupling and modular forms in N = 2 string models with a Wilson line, ” Nucl. Phys., B491, 147-183 (1997); hep-th/9608154.Google Scholar
  9. 9.
    J. H. Conway, “The automorphism group of the 26 dimensional even Lorentzian lattice, ” J. Algebra, 80, 159-163 (1983).Google Scholar
  10. 10.
    R. Dijkgraaf,E. Verlinde, andH. Verlinde, “Counting dyons in N = 4 string theory, ” Nucl. Phys., B484, 543-561 (1997); hep-th/9607026.Google Scholar
  11. 11.
    V. A. Gritsenko, “Modular forms and moduli spaces of Abelian and K3 surfaces, ” Algebra Analiz, 6:6, 65-102 (1994).Google Scholar
  12. 12.
    V. A. Gritsenko, “Jacobi functions of n variables, ” Zap. Nauchn. Sem. LOMI, 168, 32-45 (1988).Google Scholar
  13. 13.
    V. A. Gritsenko, “Arithmetical lifting and its applications, ” In: Number Theory. Proceedings of Paris Seminar 1992-93 (S. David, ed.), Cambridge Univ. Press (1995), pp. 103-126.Google Scholar
  14. 14.
    V. A. Gritsenko, “Irrationality of the moduli spaces of polarized abelian surfaces, ” Intern. Math. Research Notices, 6, 235-243 (1994); in full form in Abelian Varieties. Proc. Eglo.stein Conf. (1993), de Gruyter, Berlin (1995), pp. 63-81.Google Scholar
  15. 15.
    V. A. Gritsenko andV. V. Nikulin, “Siegel automorphic form correction of some Lorentzian Kac-Moody Lie algebras, ” Amer. J. Math., 119, No. 1, 181-224 (1997); alg-geom/9504006.Google Scholar
  16. 16.
    V. A. Gritsenko andV. V. Nikulin, “Siegel automorphic form correction of a Lorentzian Kac-Moody algebra, ” C. R. Acad. Sci. Paris, Sér. A-B, 321, 1151-1156 (1995).Google Scholar
  17. 17.
    V. A. Gritsenko andV. V. Nikulin, “K3 surfaces, Lorentzian Kac-Moody algebras, and mirror symmetry, ” Math. Res. Lett., 3, No. 2, 211-229 (1996); alg-geom/9510008.Google Scholar
  18. 18.
    V. A. Gritsenko andV. V. Nikulin, “The Igusa modular forms and 'the simplest' Lorentzian Kac-Moody algebras, ” Mat. Sb., 187, No. 11, 27-66 (1996); alg-geom/9603010.Google Scholar
  19. 19.
    V. A. Gritsenko andV. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras, I, ” Intern. J. Math., 9, No. 2, 153-199 (1998); alg-geom/9610022.Google Scholar
  20. 20.
    V. A. Gritsenko andV. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras, II, ” Intern. J. Math., 9, No. 2, 201-275 (1998); alg-geom/9611028.Google Scholar
  21. 21.
    V. A. Gritsenko andV. V. Nikulin, A lecture on arithmetic mirror symmetry and Calabi-Yau manifolds, Duke e-prints alg-geom/961 2002 (1996).Google Scholar
  22. 22.
    J. Harvey andG. Moore, “Algebras, BPS-states, and strings, ” Nucl. Phys., B463, 315-368 (1996); hep-th/9510182.Google Scholar
  23. 23.
    J. Harvey andG. Moore, On the algebras of BPS-states, Preprint hep-th/9609017 (1996).Google Scholar
  24. 24.
    J. Igusa, “On Siegel modular forms of genus two, II, ” Amer. J. Math., 84, No. 2, 392-412 (1964).Google Scholar
  25. 25.
    V. Kac, Infinite-Dimensional Lie Algebras, Cambridge Univ. Press (1990).Google Scholar
  26. 26.
    T. Kawai, “N = 2 heterotic string threshold correction, K3 surfaces and generalized Kac-Moody super algebra, ” Phys. Lett., B372, 59-64 (1996); hep-th/9512046.Google Scholar
  27. 27.
    T. Kawai, “String duality and modular forms, ” Phys. Lett., B397, 51-62 (1997); hep-th/9607078.Google Scholar
  28. 28.
    G. Moore, “String duality, automorphic forms, and generalized Kac-Moody algebras, ” Nucl. Phys. Proc. Suppl., 67, 56-67 (1998); hep-th/9710198.Google Scholar
  29. 29.
    V. V. Nikulin, “Integral symmetric bilinear forms and some of their geometric applications, ” Izv. Akad. Nauk SSSR, Ser. Mat., 43, 111-177 (1979).Google Scholar
  30. 30.
    V. V. Nikulin, “On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections. Algebraic-geometric applications, ” In: Itogi Nauki i Tekhn. Sovremennye Problemy Matematiki, All-Russian Institute for Scientific and Technical Information (VINITI), Moscow (1981), pp. 3-114.Google Scholar
  31. 31.
    V. V. Nikulin, “On arithmetic groups generated by reflections in Lobachevsky spaces, ” Izv. Akad. Nauk SSSR, Ser. Mat., 44, 637-669 (1980).Google Scholar
  32. 32.
    V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces, ” Izv. Akad. Nauk SSSR Ser. Mat., 45, No. 1, 113-142 (1981).Google Scholar
  33. 33.
    V. V. Nikulin, “Surfaces of type K3 with finite automorphism group and Picard group of rank three, ” Tr. Mat. Inst. Rossiisk. Akad. Nauk, 165, 113-142 (1984).Google Scholar
  34. 34.
    V. V. Nikulin, “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, ” In: Proc. Int. Congr. Math. Berkeley, Vol. 1 (1986), pp. 654-669.Google Scholar
  35. 35.
    V. V. Nikulin, A lecture on Kac-Moody Lie algebras of the arithmetic type, Preprint # 1994-16, Queen's University, Canada (1994); alg-geom/9412003.Google Scholar
  36. 36.
    V. V. Nikulin, “Reflection groups in Lobachevsky spaces and the denominator identity for Lorentzian Kac-Moody algebras, ” Izv. Rossiisk. Akad. Nauk, Ser. Mat., 60, No. 2, 73-106 (1996); alggeom/ 9503003.Google Scholar
  37. 37.
    V. V. Nikulin, “The remark on discriminants of K3 surfaces moduli as sets of zeros of automorphic forms, ” J. Math. Sci., 81, No. 3, 2738-2743 (1996); alg-geom/9512018.Google Scholar
  38. 38.
    L. S. Pontryagin, Topological Groups, Gordon and Breach (1966).Google Scholar
  39. 39.
    U. Ray, “A character formula for generalized Kac-Moody superalgebras, ” J. Algebra, 177, 154-163 (1995).Google Scholar
  40. 40.
    E. B. Vinberg, “The absence of crystallographic reflection groups in Lobachevsky spaces of large dimension, ” Tr. Mosk. Mat. Obshch., 47, 68-102 (1984).Google Scholar

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© Plenum Publishing Corporation 2001

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