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Mat. Sb., 1996, Volume 187, Number 11, Pages 27–66 (Mi msb171)  

This article is cited in 32 scientific papers (total in 32 papers)

Igusa modular forms and 'the simplest' Lorentzian Kac–Moody algebras

V. A. Gritsenkoa, V. V. Nikulinb

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Automorphic corrections for the Lorentzian Kac–Moody algebras with the simplest generalized Cartan matrices of rank 3,
$$ A_{1,0}=\begin{pmatrix} \hphantom{-}{2}&\hphantom{-}{0}&{-1}
\hphantom{-}{0}&\hphantom {-}{2}&{-2}
{-1}&{-2}&\hphantom {-}{2} \end{pmatrix} \quadand\quad A_{1,\mathrm {I}}=\begin {pmatrix} \hphantom {-}{2}&{-2}&{-1}
{-2}&\hphantom {-}{2}&{-1}
{-1}&{-1}&\hphantom {-}{2} \end{pmatrix} $$
are found. For $A_1,0$ this correction, which is a generalized Kac–Moody Lie super algebra, is delivered by $\chi_{35}(Z)$, the Igusa $\operatorname{Sp}_4(\mathbb Z)$-modular form of weight $35$, while for $A_{1,\mathrm{I}}$ it is given by some Siegel modular form $\widetilde \Delta_{30}(Z)$ of weight 30 with respect to a 2-congruence subgroup of $\operatorname{Sp}_4(\mathbb Z)$. Expansions of $\chi_{35}(Z)$ and $\widetilde\Delta_{30}(Z)$ in infinite products are obtained and the multiplicities of all the roots of the corresponding generalized Lorentzian Kac–Moody superalgebras are calculated. These multiplicities are determined by the Fourier coefficients of certain Jacobi forms of weight 0 and index 1.
The method adopted for constructing $\chi_{35}(Z)$ and $\widetilde\Delta_{30}(Z)$ leads in a natural way to an explicit construction (as infinite products or sums) of Siegel modular forms whose divisors are Humbert surfaces with fixed discriminants. A geometric construction of these forms was proposed by van der Geer in 1982.
To show the prospects for further studies, the list of all hyperbolic symmetric generalized Cartan matrices $A$ with the following properties is presented: $A$ is a matrix of rank 3 and of elliptic or parabolic type, has a lattice Weyl vector, and contains a parabolic submatrix $\widetilde{\mathbb A}_1$.

DOI: https://doi.org/10.4213/sm171

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English version:
Sbornik: Mathematics, 1996, 187:11, 1601–1641

Bibliographic databases:

UDC: 512.818.4+512.817.72+511.334+512.774
MSC: Primary 17B67, 17B70, 11F46; Secondary 14J15, 14J28
Received: 04.06.1996

Citation: V. A. Gritsenko, V. V. Nikulin, “Igusa modular forms and 'the simplest' Lorentzian Kac–Moody algebras”, Mat. Sb., 187:11 (1996), 27–66; Sb. Math., 187:11 (1996), 1601–1641

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Gritsenko, VA, “Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras”, American Journal of Mathematics, 119:1 (1997), 181  crossref  mathscinet  zmath  isi  elib
    2. Gritsenko, VA, “Automorphic forms and Lorentzian Kac-Moody algebras. Part II”, International Journal of Mathematics, 9:2 (1998), 201  crossref  mathscinet  zmath  isi  elib
    3. Gritsenko, VA, “Automorphic forms and Lorentzian Kac-Moody algebras. Part I”, International Journal of Mathematics, 9:2 (1998), 153  crossref  mathscinet  zmath  isi  elib
    4. V. V. Nikulin, “On the Classification of Hyperbolic Root Systems of Rank Three”, Proc. Steklov Inst. Math., 230:3 (2000), 1–241  mathnet  mathscinet  zmath
    5. Ray, U, “Generalized Kac-Moody algebras and some related topics”, Bulletin of the American Mathematical Society, 38:1 (2000), 1  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    6. Nikulin, VV, “A remark on algebraic surfaces with polyhedral Mori cone”, Nagoya Mathematical Journal, 157 (2000), 73  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Gritsenko, VA, “The arithmetic mirror symmetry and Calabi-Yau manifolds”, Communications in Mathematical Physics, 210:1 (2000), 1  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    8. V. A. Gritsenko, V. V. Nikulin, “On classification of Lorentzian Kac–Moody algebras”, Russian Math. Surveys, 57:5 (2002), 921–979  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Dern, T, “Graded rings of Hermitian modular forms of degree 2”, Manuscripta Mathematica, 110:2 (2003), 251  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. Guerzhoy, P, “On the Hecke equivariance of the Borcherds isomorphism”, Bulletin of the London Mathematical Society, 38 (2006), 93  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    11. Dabholkar A., Narnpuri S., “Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift”, Journal of High Energy Physics, 2007, no. 11, 077  crossref  mathscinet  zmath  isi  scopus  scopus
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    16. H.H.. Kim, Kyu-Hwan Lee, “Automorphic correction of the hyperbolic Kac-Moody algebra E10”, J. Math. Phys, 54:9 (2013), 091701  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    17. Caner Nazaroglu, “Jacobi forms of higher index and paramodular groups in $ \mathcal{N} $ = 2, D = 4 compactifications of string theory”, J. High Energ. Phys, 2013:12 (2013)  crossref  mathscinet  isi  scopus  scopus
    18. Kim H.H., Lee K.-H., “Root Multiplicities of Hyperbolic Kac-Moody Algebras and Fourier Coefficients of Modular Forms”, Ramanujan J., 32:3 (2013), 329–352  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Persson D., Volpato R., “Second-Quantized Mathieu Moonshine”, Commun. Number Theory Phys., 8:3 (2014), 403–509  crossref  mathscinet  zmath  isi  scopus  scopus
    20. Kim H.H., Lee K.-H., “Rank 2 Symmetric Hyperbolic Kac-Moody Algebras and Hilbert Modular Forms”, J. Algebra, 407 (2014), 81–104  crossref  mathscinet  zmath  isi  scopus  scopus
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