Rota-baxter algebras

Let A be an algebra over the ring R . The linear operation P:A→A satisfying the following identity for some λ ∈ R and for all x,y ∈A P(x)∙P(y)=P(P(x)∙y)+P(x∙P(y) )+λP(x∙y) then (A,P) is a Rota-Baxter algebra of weight λ and P is a Rota-Baxter operator. In this thesis, Rota-Baxter algebras are described, the struction of free Baxter algebras is investigated, Gröbner-Shirshov basis for Rota-Baxter algebras is given, and it is shown how a free Lie algebra with Rota-Baxter operator can be constructed.