On simple compact Lie skew braces

TL;DR

This paper investigates the simplicity of compact connected Lie skew braces, establishing conditions under which they are trivial or nearly trivial, and contrasting with noncompact examples that show more complexity.

Contribution

It characterizes the simplicity of compact connected Lie skew braces, proving they are trivial or almost trivial unless in specific noncompact cases.

Findings

Any compact connected simple Lie skew brace is either trivial on S^1 or both Lie groups are simple.

Every connected compact solvable Lie skew brace is trivial.

A noncompact example shows that the rigidity does not extend beyond compact cases.

Abstract

We study simplicity of Lie skew braces from both global and infinitesimal perspectives. After reviewing the correspondence between connected Lie skew braces, simply transitive affine actions, and post-Lie algebras, we investigate ideals and rigidity phenomena. Our main result concerns compact connected Lie skew braces. We prove that any compact connected simple Lie skew brace is either the trivial Lie skew brace on \(S^1\), or both of its underlying Lie groups are simple and the brace is trivial or almost trivial. Consequently, apart from the exceptional \(S^1\) case, simplicity of a compact connected Lie skew brace is equivalent to simplicity of either underlying Lie group. We also show that every connected compact solvable Lie skew brace is trivial. Finally, we construct a noncompact example demonstrating that this rigidity phenomenon does not hold in general: there exists a connected simply connected simple Lie skew brace whose additive and multiplicative Lie groups are both solvable.