Gröbner-Shirshov bases for commutative algebras with multiple operators and free commutative Rota-Baxter algebras. (Q2920908)

Let \(K\) be a unitary commutative ring. An associative algebra with multiple operators is an associative \(K\)-algebra with a set of multi-linear operators. A Rota-Baxter algebra of weight \(\lambda\) (\(\lambda\in K\)) is an associative \(K\)-algebra \(R\) with a linear operator \(P\colon R\to R\) satisfying the Rota-Baxter identity: \(P(u)P(v)=P(uP(v))+P(P(u)v)+\lambda P(uv)\). In this paper, the Composition-Diamond lemma for commutative associative algebras with multiple operators is established. As applications, Gröbner-Shirshov bases and linear bases of free commutative Rota-Baxter algebra, free commutative \(\lambda\)-differential algebra and free commutative \(\lambda\)-differential Rota-Baxter algebra are given, respectively.