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

TL;DR

This paper investigates the rigidity of pairs of Lie algebras with a post-Lie algebra structure, proving that semisimple cases are rigid and establishing new existence results for other algebra pairs.

Contribution

It proves that semisimple Lie algebras in such pairs must be isomorphic, and provides new existence results for pairs involving complete, reductive, solvable, or nilpotent Lie algebras.

Findings

Semisimple Lie algebras in pairs with a post-Lie structure are necessarily isomorphic.

Rigidity results for pairs where rak{g} is semisimple.

Existence results for pairs with rak{g} complete or reductive, and rak{n} solvable or nilpotent.

Abstract

We study rigidity questions for pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $\mathfrak{g}$ is semisimple and $\mathfrak{n}$ is arbitrary, then we have rigidity in the sense that $\mathfrak{g}$ and $\mathfrak{n}$ must be isomorphic. The proof uses a result on the decomposition of a Lie algebra $\mathfrak{g}=\mathfrak{s}_1\dotplus \mathfrak{s}_2$ as the direct vector space sum of two semisimple subalgebras. We show that $\mathfrak{g}$ must be semisimple and hence isomorphic to the direct Lie algebra sum $\mathfrak{g}\cong \mathfrak{s}_1\oplus \mathfrak{s}_2$. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$. We prove additional existence results for pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{g}$ is complete, and for pairs, where $\mathfrak{g}$ is reductive with $1$-dimensional center and $\mathfrak{n}$ is solvable or nilpotent.