2010/03/02 by Lisa Carbone, Sjuvon Chung, Leigh Cobbs +4 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Dynkin diagram #Group (periodic table) #Lie algebra #Linguistics #Mathematics #Philosophy #Physics #Pure mathematics #Quantum mechanics #Root (linguistics) #Weyl group #hep-th #math-ph #math.MP #math.RT
paper · pdf · doi:10.1088/1751-8113/43/15/155209
J. Phys. A: Math. Theor (to appear)
arxiv created 2010/03/02 · openalex publication_date 2010/03/26 · arxiv updated 2015/05/18 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Saçlio lu, we present the 238 hyperbolic Dynkin diagrams in ranks 3–10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.