On Simply Connected Simple Lie Skew Braces with Nilpotent Multiplicative Group

Abstract

We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if $(G,\cdot,\circ)$ is a simply connected Lie skew brace with nilpotent multiplicative Lie group and $\dim G>1$, then $(G,\cdot,\circ)$ is not simple. Thus, in the simply connected Lie setting, nilpotency of the multiplicative group is incompatible with simplicity in every dimension greater than one. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show that trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals.