A classification of subsystems of a root system

TL;DR

This paper classifies root system homomorphisms that preserve Cartan integers, exploring their isomorphism classes and automorphisms, and introduces the concept of dual pairs to analyze outer automorphisms.

Contribution

It provides a comprehensive classification of root system homomorphisms and introduces dual pairs to study automorphism actions, advancing understanding of root system structures.

Findings

Classified isomorphic classes of root system homomorphisms.

Analyzed the impact of Weyl group and automorphisms on these classes.

Introduced the concept of dual pairs for root systems.

Abstract

We classify isomorphic classes of the homomorphisms of a root system $\Xi$ to a root system $\Sigma$ which do not change Cartan integers. We examine several types of isomorphic classes defined by the Weyl group of $\Sigma$, that of $\Xi$ and the automorphisms of $\Sigma$ or $\Xi$ etc. We also distinguish the subsystem generated by a subset of a fundamental system. We introduce the concept of the dual pair for root systems which helps to study the action of the outer automorphism of $\Xi$ on the homomorphisms.