Problems in Lie Group Theory

TL;DR

This paper reviews the historical development of Lie group theory, highlights established results, and discusses unresolved questions in the field, especially concerning exceptional groups and torsion.

Contribution

It provides a historical overview and raises open questions about Lie groups, particularly focusing on unresolved issues in exceptional groups and torsion phenomena.

Findings

Historical development of Lie groups summarized

Key unresolved questions identified in Lie group theory

Discussion on torsion in exceptional groups

Abstract

The theory of Lie groups and representations was developed by Lie, Killing, Cartan, Weyl and others to a degree of quasi-perfection, in the years 1870-1930 >. The main topological features of compact simple Lie groups were elucidated in the 40s by H. Hopf, Pontriagin and others. The exceptional groups were studied by Chevalley, Borel, Freudenthal etc. in 1949-1957. Torsion in the exceptional groups was considered by Toda, Adams etc. in the 80s. However, one can still ask some questions for which the answer is either incomplete or absent, at least to this speaker. We would like to raise and discuss some of them in this communication.