On symmetric closed subsets of real affine root systems

TL;DR

This paper investigates the conditions under which symmetric closed subsets of real affine root systems are subroot systems, providing explicit descriptions and exploring their connection to regular subalgebras.

Contribution

It characterizes when symmetric closed subsets of real affine root systems are subroot systems and offers explicit descriptions, extending known results from finite to affine cases.

Findings

Identifies conditions for symmetric closed subsets to be subroot systems

Provides explicit descriptions of symmetric closed subsets

Explores the link to regular subalgebras

Abstract

Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit description of symmetric closed subsets of real affine root systems. At the end, using our results, we study the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them.