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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 5(347), Pages 79–138 (Mi umn553)  

This article is cited in 25 scientific papers (total in 25 papers)

On classification of Lorentzian Kac–Moody algebras

V. A. Gritsenkoab, V. V. Nikulincd

a University of Sciences and Technologies
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
c Steklov Mathematical Institute, Russian Academy of Sciences
d University of Liverpool

Abstract: The general theory of Lorentzian Kac–Moody algebras is considered. This theory must serve as a hyperbolic analogue of the classical theories of finite-dimensional semisimple Lie algebras and affine Kac–Moody algebras. The first examples of Lorentzian Kac–Moody algebras were found by Borcherds. Here general finiteness results for the set of Lorentzian Kac–Moody algebras of rank $\geqslant 3$ are considered along with the classification problem for these algebras. As an example, a classification is given for Lorentzian Kac–Moody algebras of rank 3 with hyperbolic root lattice $S_t^*$, symmetry lattice $L_t^*$, and symmetry group $\widehat O^+(L_t)$, $t\in\mathbb N$, where $S_t$ and $L_t$ are given by
\begin{gather*} S_t=H\oplus\langle 2t\rangle=(\begin{smallmatrix}0&0&-1
0&2t&0\1&0&0\end{smallmatrix}), \quad L_t=H\oplus S_t=(\begin{smallmatrix}0&0&0&0&-1
0&0&0&-1&0
0&0&2t&0&0
0&-1&0&0&0\1&0&0&0&0\end{smallmatrix}),
H=(\begin{smallmatrix}0&-1\1&0\end{smallmatrix}), \quad \end{gather*}
and $\widehat O^+(L_t)=\{g\in O^+(L_t)\mid g$ is trivial on $L_t^*/L_t\}$, is the extended paramodular group. This is perhaps the first example in which a large class of Lorentzian Kac–Moody algebras has been classified.

DOI: https://doi.org/10.4213/rm553

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English version:
Russian Mathematical Surveys, 2002, 57:5, 921–979

Bibliographic databases:

UDC: 512.818.4+512.817.72+511.334+512.774
MSC: Primary 17B67; Secondary 11F22, 11F50, 14J15, 14J28, 81R10
Received: 17.01.2002

Citation: V. A. Gritsenko, V. V. Nikulin, “On classification of Lorentzian Kac–Moody algebras”, Uspekhi Mat. Nauk, 57:5(347) (2002), 79–138; Russian Math. Surveys, 57:5 (2002), 921–979

Citation in format AMSBIB
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\paper On classification of Lorentzian Kac--Moody algebras
\jour Uspekhi Mat. Nauk
\yr 2002
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\pages 79--138
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    18. Allcock D., “Root Systems For Lorentzian Kac-Moody Algebras in Rank 3”, 47, no. 2, 2015, 325–342  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475  crossref  mathscinet  zmath  isi  elib  scopus
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