Maximal abelian subgroups of compact simple Lie groups

TL;DR

This paper classifies maximal abelian subgroups of compact simple Lie groups and their automorphism groups, providing a comprehensive understanding of their structure and associated gradings.

Contribution

It introduces a classification of abelian subgroups with specific centralizer properties, leading to a complete classification of maximal abelian subgroups and fine gradings.

Findings

Classification of abelian subgroups with equal centralizer and subgroup dimensions

Complete classification of maximal abelian subgroups of compact simple groups

Enumeration of fine group gradings of complex simple Lie algebras

Abstract

We classify abelian subgroups of the automorphism group of any compact simple Lie algebra whose centralizer has the same dimension as the dimension of the subgroup. This leads to a classification of the maximal abelian subgroups of compact simple groups of adjoint type and a classification of the fine group gradings of complex simple Lie algebras.