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

摘要

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 id="ilm0017">n=sl2(C) was studied. In this paper, we determine all Lie algebras id="ilm0018">g such that there exists a post-Lie algebra structure on id="ilm0019">(g,n) with id="ilm0020">n=sl2(C)⊕sl2(C).