The topology of a semisimple Lie group is essentially unique

TL;DR

This paper proves that the topology of a semisimple Lie group is essentially unique, showing that any isomorphism with a locally compact group is automatically a homeomorphism, highlighting the rigidity of such groups.

Contribution

It establishes the topological rigidity of semisimple Lie groups, demonstrating that abstract isomorphisms are necessarily topological under certain conditions.

Findings

Any abstract isomorphism between a semisimple Lie group and a locally compact group is a homeomorphism.

The rigidity holds for absolutely simple groups, with considerations for complex groups involving field automorphisms.

The topology of semisimple Lie groups is essentially unique, up to isomorphism.

Abstract

We study locally compact group topologies on semisimple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.