Structural and rigidity properties of Lie skew braces

TL;DR

This paper studies the structural and rigidity properties of Lie skew braces, revealing how their algebraic features relate and providing existence results across different Lie group classes.

Contribution

It introduces new rigidity and flexibility results for Lie skew braces, connecting their properties with Lie group classifications and providing explicit construction methods.

Findings

Linearity and solvability transfer from one group law to the other in connected LSBs.

Nilpotent or semisimple extit{(G, extcircled{ } )} impose solvability or isomorphism on extit{(G, extperiodcentered)}.

Complete existence table for non-trivial LSBs across six standard Lie-group classes.

Abstract

We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same identity element. LSBs extend skew left braces, which are central to the study of non-involutive set-theoretic solutions of the Yang--Baxter equation, to the smooth category. Our first main result shows that, for every connected LSB $(G,\cdot,\circ)$, linearity (in the simply-connected case) and solvability carry over from $(G,\cdot)$ to $(G,\circ)$, whereas the converse direction is rigid: if $(G,\circ)$ is nilpotent (respectively, semisimple) then $(G,\cdot)$ is forced to be solvable (respectively, isomorphic to $(G,\circ)$). Our second results provides two ``flexibility'' statements: every non-linear simply connected Lie group \((G, \cdot )\) admits an LSB \((G,\cdot,\circ)\) such that \((G,\circ)\) is linear, and every simply connected solvable Lie group \((G, \circ )\) supports a LSB \((G,\cdot,\circ)\) such that \((G,\cdot)\) is nilpotent. A third result provides a complete existence table for non-trivial LSBs across the six standard Lie-group classes, abelian, nilpotent (non-abelian), solvable (non-nilpotent), simple, semisimple (non-simple) and mixed type, identifying precisely when a LSB can be built and when only the trivial or no structure occurs. Both the explicit constructions and the properties established in our theorems rely on a factorisation technique for Lie groups, on the correspondence between LSBs and regular subgroups of the affine group $\operatorname{Aff}(G,\cdot)$, which renders LSB theory equivalent to simply transitive affine actions, and on the theory of post-Lie algebras together with their integrability properties.