On the commutator length of compact Lie groups

TL;DR

This paper proves that in compact Lie groups, the identity component of the derived subgroup consists solely of commutators, and explores implications for the homotopy type of classifying spaces related to commutativity.

Contribution

It establishes a fundamental property of the derived subgroup in compact Lie groups and applies this to analyze the homotopy type of classifying spaces for commutativity.

Findings

The identity component of the derived subgroup is exactly the set of commutators.

Application to the homotopy type of classifying spaces for commutativity.

Provides new insights into the structure of compact Lie groups.

Abstract

In this short note we show that the path-connected component of the identity of the derived subgroup of a compact Lie group consists just of commutators. We also discuss an application of our main result to the homotopy type of the classifying space for commutativity for a compact Lie group whose path-connected component of the identity is abelian.