Classification of Simple Lie Algebras on a Lattice

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

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