Classification of Simple Lie Algebras on a Lattice

Let $\mathrm{\Lambda }$ be a lattice of rank $n$. A Lie algebra on the lattice $\mathrm{\Lambda }$ is a Lie algebra ${\cal L}=\oplus_{\lambda\in\Lambda},{\cal L}{\lambda}$suchthat$\dim,{\cal L}\lambda=1$forall$\lambda$. In this article, we classify all simple graded Lie algebras on a lattice.