A new approach to Rota–Baxter coalgebras
Rota–Baxter algebras are a useful tool applied in many branches of mathematics. However, it is difficult to construct examples of Rota–Baxter coalgebras. In this paper, two new approaches to Rota–Baxter coalgebras of weight $-1$ are introduced, via Hopf coquasigroup theory and Hopf $\pi $-algebra theory. In order to do so, the notions of Rota–Baxter linear equation system, Hopf quasicomodule coalgebra, and $H_p$-Hopf module coalgebra are introduced and discussed. We present numerous new examples of Rota–Baxter coalgebras. Our ideas of constructing such examples may be viewed as a guide to further development.