Classification of Simple Lie Algebras on a Lattice

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