Matching Rota-Baxter Systems and Gröbner-Shirshov Bases

Motivated by the concept of matching Rota-Baxter algebras arising from polarized associative Yang-Baxter equations and Volterra integral equations, we introduce the notion of a matching Rota-Baxter system, which generalizes the Rota-Baxter system proposed by Brzeziński. We show that this notion is also related to Yang-Baxter pairs and to matching pre-Lie algebras. We then modify the definition of matching Rota-Baxter systems by adding a curvature term, and make a connection with matching pre-Lie algebras and with compatible associative algebras. Furthermore, we study matching Rota-Baxter systems on a dendriform algebra and show how they induce matching quadri-algebra structures. Finally, we give a linear basis of free matching Rota-Baxter system by Gröbner-Shirshov bases methods.