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 Mₙ(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 rb(A) as the nilpotency index for Rota-Baxter operators of weight zero on A. We show that rb(Mₙ(F)) = 2n-1 provided that characteristic of F is zero.