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arXiv · 2510.19703

Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences

Kai Neergård
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Abstract

A novel method of determining which Dynkin diagrams represent simple finite-dimensional Lie algebras over $\mathbb{C}$ is presented. It is based on a condition that is both necessary and sufficient for a suitably defined Cartan matrix to be expressible by scalar products in a Euclidean vector space. The sufficiency of this condition makes unnecessary subsequent verification of the existence of a Lie algebra or root system corresponding to each Dynkin diagram by explicit construction. The Dynkin diagrams are selected by examination of an easily calculated sequence of minors of a symmetrised Cartan matrix. These minors are mostly integers.

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Kai Neergård. 2025-10-22. Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences. https://arxiv.org/abs/2510.19703

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