Injective Rota–Baxter Operators of Weight Zero on F[x]

Abstract

We describe all Rota Baxter operators R of weight zero on the algebra U 3 (F ) of upper-triangular matrices of order three over a field of characteristic 0. For this, we apply the following three ingredients: properties of R(1), conjugation with suitable (anti)automorphisms of U 3 (F ), and computation with the help of Singular of the Gröbner basis for the system of the equations arisen on the coefficients of R.