Rota–Baxter operators and post-Lie algebra structures on semisimple Lie algebras D. Burde and V. Gubarev Communications in Algebra®

Abstract

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 ( C ) 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 ( C ) ⊕ s l 2 ( C ).