Small commutators in compact semisimple Lie groups and Lie algebras

TL;DR

This paper proves that in compact semisimple Lie groups and Lie algebras, the commutator map sends neighborhoods of the identity to neighborhoods of the identity, revealing a local surjectivity property.

Contribution

It establishes a local surjectivity result for the commutator map in compact semisimple Lie groups and Lie algebras, a new insight into their structure.

Findings

Commutator map is locally surjective near the identity.

Neighborhoods of the identity are mapped to neighborhoods of the identity.

The result applies to both Lie groups and Lie algebras.

Abstract

We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.