Rota-Baxter operators on BiHom-associative algebras and related structures. (English) Zbl 1472.17097
Following the work of Makhlouf and Yau on Rota-Baxter Hom-algebras, Rota-Baxter on BiHom associative algebras are introduced. A BiHom associative algebra is an associative algebra with two commuting algebra endomorphisms \(\alpha\) and \(\beta\), with the following axiom:
\[
\alpha(x)yz=xy\beta(z).
\]
Firstly, BiHom dendriform, Zinbiel, tridendriform and quadri algebras are introduced, and classical relations between these objects are extended to the BiHom context. It is also proved that a Yau twist exists for all of them. Secondly, a theory of Rota-Baxter operators on these objects is developed. It is proved that if the Rota-Baxter operator \(R\) commutes with both algebra endomorphisms, then the Yau twist is also a Rota-Baxter algebra. Free Rota-Baxter BiHom associative algebras are described with the help of planar trees and with considerations of functors related to Rota-Baxter structures. The paper ends with considerations on weak pseudotwistors.
Reviewer: Loïc Foissy (Calais)
MSC:
| 17D30 | (non-Lie) Hom algebras and topics |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 15A04 | Linear transformations, semilinear transformations |
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