Rota–Baxter cosystems and coquasitriangular mixed bialgebras

Abstract

In this paper, we present a dual version of T. Brzeziński’s results about Rota–Baxter systems which appeared in [Rota–Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra 460 (2016) 1–25]. Then as a generalization to bialgebras, we introduce the notion of Rota–Baxter bisystem and construct various examples of Rota–Baxter bialgebras and bisystems in dimensions 2, 3 and 4. On the other hand, we introduce a new type of bialgebras (named mixed bialgebras) which consist of an associative algebra and a coassociative coalgebra satisfying the compatible condition determined by two coderivations. We investigate coquasitriangular mixed bialgebras and the particular case of coquasitriangular infinitesimal bialgebras, where we give the double construction. Also, we show in some cases that Rota–Baxter cosystems can be obtained from a coquasitriangular mixed bialgebras.