Structural and rigidity properties of Lie skew braces

Damele, Marco;Loi, Andrea

2026-01-01

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Abstract

We investigate structural and rigidity properties of Lie skew braces (LSBs), objects essentially known in the literature as 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 (Theorem 1.1) shows that, for every connected LSB (G,⋅,∘), linearity (in the simply-connected case) and solvability carry over from (G,⋅) to (G,∘), whereas the converse direction is rigid: if (G,∘) is nilpotent (respectively, semisimple) then (G,⋅) is forced to be solvable (respectively, isomorphic to (G,∘)). Theorem 1.3 provides two “flexibility” statements: every non-linear simply connected Lie group (G,⋅) admits an LSB (G,⋅,∘) such that (G,∘) is linear, and every simply connected solvable Lie group (G,∘) supports an LSB (G,⋅,∘) such that (G,⋅) is nilpotent. A third result (Theorem 1.4) 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 an 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 Aff(G,⋅), which renders LSB theory equivalent to simply transitive affine actions, and on the theory of post–Lie algebras together with their integrability properties.