On classification of Lorentzian Kac–Moody algebras

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=\left(\begin{smallmatrix}0&0&-1\\0&2t&0\\-1&0&0\end{smallmatrix}\right), \quad L_t=H\oplus S_t=\left(\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}\right), \\ H=\left(\begin{smallmatrix}0&-1\\-1&0\end{smallmatrix}\right), \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.