On ${\pi}$-systems of symmetrizable Kac-Moody algebras

TL;DR

This paper investigates the structure of $\pi$-systems in symmetrizable Kac-Moody algebras, establishing a partial order and methods for constructing and classifying these systems, with applications to hyperbolic Dynkin diagrams.

Contribution

It introduces a partial order on $\pi$-systems, provides principles for their construction, and classifies maximal hyperbolic Dynkin diagrams in certain ranks.

Findings

Morita's relation defines a partial order on $\\pi$-systems of specific Kac-Moody types.

Develops principles for constructing $\\pi$-systems and identifying forbidden diagrams.

Determines maximal hyperbolic Dynkin diagrams in ranks 3-10 relative to the Morita order.

Abstract

Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $\pi$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of $\mathfrack{g}$, and were originally studied by Dynkin, Morita and Naito. We show that the binary relation introduced by Morita defines a partial order on the set of $\mathfrack{g}$ of finite, untwisted affine or hyperbolic type. We also formulate general principles for constructing $\pi$-systems as well as for finding forbidden diagrams that cannot occur as Dynkin diagrams of $\pi$-systems of a given $\mathfrack{g}$. Among other applications, we use this to determine the set of maximal hyperbolic Dynkin diagrams in ranks $3$-$10$ relative to the Morita partial order.