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

This article is cited in 34 scientific papers (total in 34 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 $\widetilde{\mathbb A}_1$.


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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
\by V.~A.~Gritsenko, V.~V.~Nikulin
\paper Igusa modular forms and 'the~simplest' Lorentzian Kac--Moody algebras
\jour Mat. Sb.
\yr 1996
\vol 187
\issue 11
\pages 27--66
\jour Sb. Math.
\yr 1996
\vol 187
\issue 11
\pages 1601--1641

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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
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    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
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    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
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