Structural and rigidity properties of Lie skew braces
Marco Damele, Andrea Loi
TL;DR
This work develops a unified framework for Lie skew braces by exploiting their equivalence with post-Lie groups and the affine-action viewpoint. It proves that for connected LSBs, linearity and solvability transfer from the $(G,\cdot)$-structure to the $(G,\circ)$-structure, while rigidity forces (G,\cdot) to be solvable if $(G,\circ)$ is nilpotent and to be isomorphic to $(G,\circ)$ if $(G,\circ)$ is semisimple. The authors also establish flexibility results showing that one can realize LSBs with opposite structures linear or nilpotent under suitable starting hypotheses, and provide a complete non-trivial existence table across six Lie-group classes, using a factorisation technique and the correspondence with simply transitive affine actions, together with post-Lie algebra integrability considerations. These results extend classifications of PLAs and illuminate when certain post-Lie structures can be integrated into LSBs, linking geometric affine actions with algebraic brace theory. The work thus advances the structural understanding and classification of LSBs and their affine-action interpretations, with consequences for the study of set-theoretic Yang–Baxter theory in the smooth category.
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.