Gr'obner-Shirshov Bases for Commutative Algebras with Multiple Operators and Free Commutative Rota-Baxter Algebras

摘要

In this paper, the Composition-Diamond lemma for commutative algebras withmultiple operators is established. As applications, the Gr\"obner-Shirshovbases and linear bases of free commutative Rota-Baxter algebra, freecommutative $\lambda$-differential algebra and free commutative$\lambda$-differential Rota-Baxter algebra are given, respectively.Consequently, these three free algebras are constructed directly by commutative$\Omega$-words.