The 2-ranks of connected compact Lie groups

TL;DR

This paper surveys the history and key results on the 2-rank of compact Lie groups, providing a complete classification for connected simple cases and highlighting their significance in mathematics.

Contribution

It offers a comprehensive survey of 2-ranks in compact Lie groups and presents the complete determination for connected simple Lie groups using 2-numbers.

Findings

Complete determination of 2-ranks for compact connected simple Lie groups

Connection of 2-ranks with important mathematical areas

Use of 2-numbers to classify 2-ranks

Abstract

The 2-rank of a compact Lie group $G$ is the maximal possible rank of the elementary 2-subgroup ${\mathbb Z}_{2}\times... {\mathbb Z}_{2}$ of $G$. The study of 2-ranks (and $p$-rank for any prime $p$) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups have been investigated by many mathematician. The 2-ranks of compact Lie groups relate closely with several important areas in mathematics. In this article, we survey important results concerning 2-ranks of compact Lie groups. In particular, we present the complete determination of 2-ranks of compact connected simple Lie groups $G$ via the 2-numbers introduced by B. Y. Chen and T. Nagano in [Un invariant g\'em\'etrique riemannien, C. R. Acad. Sci. Paris, 295 (1982), 389--391] and [A Riemannian geometric invariant and its applications to a problem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), 273--297].