Modules of polynomial Rota-Baxter Algebras and matrix equations

Abstract

The all Rota-Baxter algebra structures on the polynomial algebra R = k[x] are well known. We study the finite dimensional modules of polynomial Rota-Baxter algebras (k[x], P) or (xk[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 (k[x], P) or (xk[x], P) is equivalent to the modules over a plane k x, y /I where I is some ideal of free algebra k x, y . Furthermore, we provide the classification of modules of polynomial Rota-Baxter algebras of weight nonzero through solution to some matrix equation.