Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

TL;DR

This paper classifies hyperbolic Dynkin diagrams, provides criteria for symmetrizability, and analyzes Weyl group orbits, advancing understanding of hyperbolic Kac-Moody algebras and their root systems.

Contribution

It introduces a simple symmetrizability criterion, classifies hyperbolic Dynkin diagrams up to rank 10, and details Weyl group orbit structures for these systems.

Findings

Maximal rank of compact hyperbolic Dynkin diagrams is 5.

Maximal rank of symmetrizable hyperbolic Dynkin diagrams is 4.

Maximal number of Weyl group orbits on real roots is 4.

Abstract

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 Sa\c{c}lio\~{g}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.