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 [13] 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.