Classification of Simple Lie Algebras on a Lattice

TL;DR

This paper classifies all simple graded Lie algebras defined on a lattice, providing a comprehensive understanding of their structure based on the lattice's properties.

Contribution

It offers a complete classification of simple graded Lie algebras on lattices, a previously unresolved problem in Lie algebra theory.

Findings

Complete classification of simple graded Lie algebras on lattices

Identification of structural properties unique to lattice-based Lie algebras

Extension of known Lie algebra classifications to lattice contexts

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.