Hyperbolic subalgebras of hyperbolic Kac-Moody algebras

TL;DR

This paper classifies regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras by analyzing their Weyl groups, providing a finite algorithm for their classification and establishing subgroup relations.

Contribution

It introduces a finite algorithm to classify all regular hyperbolic subalgebras and establishes the correspondence between Weyl group subgroups and algebra subalgebras.

Findings

Classification of subgroup relations between Weyl groups.

Existence of algebra-subalgebra pairs for each subgroup relation.

Finite algorithm for classifying hyperbolic subalgebras.

Abstract

We investigate regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras via their Weyl groups. We classify all subgroups relations between Weyl groups of hyperbolic Kac-Moody algebras, and show that for every pair of a group and subgroup their exists at least one corresponding pair of algebra and subalgebra. We also present a finite algorithm classifying all regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras.