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

Rota-Baxter operators R of weight 1 on $n$ are in bijective correspondence to post-Lie algebra structures on pairs $(g,n)$ , where $n$ is complete. We use such Rota-Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras $(g,n)$ , where $n$ is semisimple. We show that for semisimple $g$ and $n$ , with $g$ or $n$ simple, the existence of a post-Lie algebra structure on such a pair $(g,n)$ implies that $g$ and $n$ are isomorphic, and hence both simple. If $n$ is semisimple, but $g$ is not, it becomes much harder to classify post-Lie algebra structures on $(g,n)$ , or even to determine the Lie algebras $g$ which can arise. Here only the case $n=s{l}_2\left(C\right)$ was studied. In this paper, we determine all Lie algebras $g$ such that there exists a post-Lie algebra structure on $(g,n)$ with $n=s{l}_2\left(C\right)\oplus s{l}_2\left(C\right)$ .