2025/10/22 by K. Neergård, Neergård, Kai
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2510.19703
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
A novel method of determining which Dynkin diagrams represent simple finite-dimensional Lie algebras over 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.