Abstract
Rota–Baxter operators R of weight 1 on are in bijective correspondence to post-Lie algebra structures on pairs , where is complete. We use such Rota–Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras , where is semisimple. We show that for semisimple and , with or simple, the existence of a post-Lie algebra structure on such a pair implies that and are isomorphic, and hence both simple. If is semisimple, but is not, it becomes much harder to classify post-Lie algebra structures on , or even to determine the Lie algebras which can arise. Here only the case was studied. In this paper, we determine all Lie algebras such that there exists a post-Lie algebra structure on with .
Highlights
Rota–Baxter operators were introduced by Baxter [3] in 1960 as a formal generalization of integration by parts for solving an analytic formula in probability theory
Rota–Baxter operators R of weight 1 on n are in bijective correspondence to post-Lie algebra structures on pairs ðg; nÞ, where n is complete
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 ðg; nÞ implies that g and n are isomorphic, and both simple
Summary
Rota–Baxter operators were introduced by Baxter [3] in 1960 as a formal generalization of integration by parts for solving an analytic formula in probability theory. It is possible (and desirable) to use results on RB-operators for the existence and classification of post-Lie algebra structures. For a complete Lie algebra n there is a bijection between PA-structures on ðg; nÞ and RB-operators of weight 1 on n. This turns out to be much more complicated than the case n 1⁄4 sl2ðCÞ, which we have done in [11]. The result we obtain shows that there are more restrictions than that
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