Modules of polynomial Rota-Baxter Algebras and matrix equations

TL;DR

This paper classifies finite-dimensional modules over polynomial Rota-Baxter algebras of nonzero weight, showing their equivalence to modules over certain free algebras and solving related matrix equations.

Contribution

It provides a novel classification of modules over polynomial Rota-Baxter algebras of nonzero weight, linking them to modules over free algebras and matrix equations.

Findings

Modules are equivalent to modules over a quotient of a free algebra.

Classification achieved through solutions to matrix equations.

Extends understanding of Rota-Baxter algebra modules beyond weight zero.

Abstract

The all Rota-Baxter algebra structures on the polynomial algebra $R={\bf k}[x]$ are well known. We study the finite dimensional modules of polynomial Rota-Baxter algebras $(\bfk[x],P)$ or $(x {\bf k} [x],P)$ of weight nonzero since some cases of weight zero have been studied. The main result shows that every module over the polynomial Rota-Baxter algebra $(\bfk[x],P)$ or $(x {\bf k} [x],P)$ is equivalent to the modules over a plane ${\bf k}\langle x,y \rangle/ I$ where $I$ is some ideal of free algebra ${\bf k}\langle x,y \rangle$. Furthermore, we provide the classification of modules of polynomial Rota-Baxter algebras of weight nonzero through solution to some matrix equation.