Rota–Baxter operators and post-Lie algebra structures on semisimple Lie algebras

Rota–Baxter operators R of weight 1 on [Image: see text] are in bijective correspondence to post-Lie algebra structures on pairs [Image: see text] , where [Image: see text] is complete. We use such Rota–Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras [Image: see text] , where [Image: see text] is semisimple. We show that for semisimple [Image: see text] and [Image: see text] , with [Image: see text] or [Image: see text] simple, the existence of a post-Lie algebra structure on such a pair [Image: see text] implies that [Image: see text] and [Image: see text] are isomorphic, and hence both simple. If [Image: see text] is semisimple, but [Image: see text] is not, it becomes much harder to classify post-Lie algebra structures on [Image: see text] , or even to determine the Lie algebras [Image: see text] which can arise. Here only the case [Image: see text] was studied. In this paper, we determine all Lie algebras [Image: see text] such that there exists a post-Lie algebra structure on [Image: see text] with [Image: see text] .