Representations of Polynomial Rota-Baxter Algebras

TL;DR

This paper explores the structure of modules over polynomial Rota-Baxter algebras, revealing their equivalence to modules over the Jordan plane and classifying Rota-Baxter modules based on indecomposable modules.

Contribution

It establishes the equivalence between modules over polynomial Rota-Baxter algebras and Jordan plane modules, extending previous decomposability results to broader fields and providing a classification scheme.

Findings

Modules over $(k[x],P)$ are equivalent to Jordan plane modules.

Extended decomposability results to fields of characteristic zero.

Classified Rota-Baxter modules up to isomorphism.

Abstract

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.