Rota–Baxter Operators on Unital Algebras

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