Rigidity results for Lie algebras admitting a post-Lie algebra structure

Abstract

We study rigidity questions for pairs of Lie algebras (g, n) admitting a post-Lie algebra structure. We show that if g is semisimple and n is arbitrary, then we have rigidity in the sense that g and n must be isomorphic. The proof uses a result on the decomposition of a Lie algebra g = s 1 ∔ s 2 as the direct vector space sum of two semisimple subalgebras. We show that g must be semisimple and hence isomorphic to the direct Lie algebra sum g ∼ = s 1 ⊕ s 2 . This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras (g, n). We prove additional existence results for pairs (g, n), where g is complete, and for pairs, where g is reductive with 1-dimensional center and n is solvable or nilpotent.