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

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 .

Decompositions of algebras and post-associative algebra structures

“…Then g is also simple and both g and n are isomorphic. This generalizes Theorem 3.1 of [6]. Then we give a new proof of Proposition 3.8 of [6], which is stated here as Corollary 4.8 and holds for arbitrary fields of characteristic zero: let (V, ·) be a post-Lie algebra structure on a pair (g, n) over a field of characteristic zero, where g is semisimple and n is complete.…”

“…We recall Proposition 2.13 in [6], see also Corollary 5.6 in [2]. The corresponding result for a post-associative algebra structure has been shown in [8], section 4 in terms of dendriform trialgebras.…”

“…In the same way one can also prove the analogous statement for post-associative algebra structures. Since every post-Lie algebra structure on (g, n), where n is complete arises by an RB-operator of weight 1 on n, Theorem 4.6 immediately implies the following corollary, which is Proposition 3.8 in [6]. Remark 4.9.…”

“…Above we have given a new proof, which also generalizes the result to arbitrary fields of characteristic zero. Below we will also give a new proof of Proposition 3.7 in [6], which relied on Proposition 3.6, too. Example 4.10.…”

“…Therefore it is very natural to use Rota-Baxter operators for the existence and classification of post-Lie algebra structures. We have already obtained several results in [6] by using Rota-Baxter operators. In this paper we obtain further results on post-Lie algebra structures and correct the proof of Proposition 3.7 and 3.8 in [6], which relied on a decomposition theorem of Lie algebras, namely Proposition 3.6 in [6], which unfortunately is in error.…”