Rota---Baxter operators on unital algebras

We state that all Rota---Baxter operators of nonzero weight on 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. For an algebra $A$, we introduce its new invariant the rb-index $\mathrm{rb}(A)$ as the nilpotency index for 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.