Representations of Polynomial Rota-Baxter Algebras

A Rota–Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota-Baxter operator. We show that studying the modules over the polynomial Rota–Baxter algebra $(k[x],P)$ is equivalent to studying the modules over the Jordan plane, and we generalize the direct decomposability results for the $(k[x],P)$-modules in [Iy] from algebraically closed fields of characteristic zero to fields of characteristic zero. Furthermore, we provide a classification of Rota–Baxter modules up to isomorphism based on indecomposable $k[x]$-modules.