Rota Baxter Operators on Truncated Polynomial Algebras

Let K be a field of characteristic zero, and let m=(x_1,...,x_n)) be a maximal ideal of the polynomial ring K[x_1,...,x_n]. We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra R=K[x_1,\dots,x_n]/m^2. For weight zero, we prove that the Rota--Baxter operators are precisely the linear maps P satisfying P^2=0 and Image(P) \subset m/m^2. For nonzero weight, a standard rescaling reduces the classification to weight one. In this case, the operators split into two disjoint families according to the value of P(1)\in{0,-1}. On the maximal ideal m/m^2, such operators induce an endomorphism L satisfying L^2 + L = 0), equivalently, -L is idempotent. We further show that each family is isomorphic to the variety of idempotent matrices.