Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di-or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di-or tri-Var-dendriform algebras is Koszul dual to the operad governing di-or trialgebras corresponding to Var ! . We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a RotaBaxter algebra of nonzero weight in Var. MSC:18D50, 17B69

We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota-Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota-Baxter operators and the solutions to the alternative Yang-Baxter equation on the Cayley-Dickson algebra. We also investigate the Rota-Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.Mathematics Subject Classification. 16T25, 17A45, 17C50.

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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 .