The isomorphism type of the centralizer of an element in a Lie group

TL;DR

This paper characterizes the structure of centralizers of elements in 1-connected simple Lie groups using minimal geodesics, and applies this to classify certain subgroups within these groups.

Contribution

It provides a novel method to determine the isomorphism type of centralizers based on minimal geodesics, aiding subgroup classification in Lie groups.

Findings

Centralizers' isomorphism types are determined by minimal geodesics.

Classification of maximal subgroups of maximal rank.

Classification of parabolic subgroups.

Abstract

Let G be an 1-connected simple Lie group, and let x\inG be a group element. We determine the isomorphism type of the centralizer C_{x} in term of a minimal geodesic joinning the group unit e\inG to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G, and the isomorphism types of parabolic subgroups of G.