Rota–Baxter systems, dendriform algebras and covariant bialgebras

Abstract

A generalisation of the notion of a Rota–Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota–Baxter equation. Rota–Baxter operators of any weights and twisted Rota–Baxter operators are solutions of the proposed system. It is shown that dendriform algebra structures of a particular kind are equivalent to Rota–Baxter systems. It is shown further that a Rota–Baxter system induces a weak pseudotwistor [Panaite and Van Oystaeyen (2015) [15]] which can be held responsible for the existence of a new associative product on the underlying algebra. Examples of solutions of Rota–Baxter systems are obtained from quasitriangular covariant bialgebras hereby introduced as a natural extension of infinitesimal bialgebras [Aguiar (2000) [3]].