Rota–Baxter operators on unital algebras

We state that all Rota–Baxter operators of nonzero weight on the Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang–Baxter equation and Rota–Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov).
We prove that all Rota–Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. We introduce a new invariant for an algebra $A$ called the RB-index $\mathrm{rb}(A)$ as the minimal nilpotency index of Rota–Baxter operators of weight zero on $A$. We show that $\mathrm{rb}(M_n(F)) = 2n-1$ provided that characteristic of $F$ is zero.