Loday-type algebras and the Rota-Baxter relation
Let \(\mathcal A\) be an associative algebra over \(\mathbb C\) or \(\mathbb R\) and let \(\lambda\) be a fixed constant. The linear operator \(R: {\mathcal A} \to {\mathcal A}\) is called Rota-Baxter operator of weight \(\lambda\) if it satisfies \(R(x)R(y)+\lambda R(xy)=R(R(x)y+xR(y)),\, x,y\in {\mathcal A}\). For \(\lambda=0\) this relation reduces to a shuffle relation. Otherwise, after normalization, one may assume that \(\lambda=1\), which, in a Lie algebraic context, is the operator form of the classical Yang-Baxter equation [see \textit{A. A. Belavin} and \textit{V. G. Drinfeld}, Triangle equations and simple Lie algebras, Classic Reviews in Mathematics and Mathematical Physics. 1. Amsterdam: Harwood Academic Publishers (1998; Zbl 0921.58073); Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 4, 93--165 (1984; Zbl 0553.58040)]. (Pay attention that the two Baxters are different persons!) In the paper under review the author considers the modified Rota-Baxter relation \(B(x)B(y)=B(B(x)y+xB(y))-xy\) satisfied by the operator \(B=1-2R\), where \(R\) is already normalized. In the Lie context of [\textit{M. A. Semenov-Tyan-Shanskij}, Funkts. Anal. Prilozh. 17, No. 4, 17--33 (1983); translation in Funct. Anal. Appl. 17, 259--272 (1983; Zbl 0535.58031)] this is the modified classical Yang-Baxter equation. The main result of the paper is that the algebras equipped with modified Rota-Baxter operators have a natural structure of dendriform trialgebras. This provides a whole class of examples of these new algebraic structures. The result is an analogue of [\textit{M. Aguiar}, Lett. Math. Phys. 54, 263--277 (2000; Zbl 1032.17038)] that algebras with Rota-Baxter operators can be considered as dendriform bialgebras. Recall that dendriform di- and trialgebras are algebras with two, respectively three binary multiplications related with some conditions for associativity. The trialgebras were introduced in [\textit{J.-L. Loday} and \textit{M. O. Ronco}, C. R. Acad. Sci., Paris, Sér. I, Math. 333, 81--86 (2001; Zbl 1010.18007)] and for dialgebras see [\textit{J.-L. Loday, A. Frabetti, F. Chapoton} respectively \textit{F. Goichot} in ``Dialgebras and related operads'', Lect. Notes Math. 1763, 7--66 (2001; Zbl 0999.17002), 67--103 (2001; Zbl 0999.17003), 105--110 (2001; Zbl 0999.17004), 111--133 (2001; Zbl 0999.17005)].
associative classical Yang-Baxter equation
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