Gröbner-Shirshov bases and linear bases for free multi-operated algebras over algebras with applications to differential Rota-Baxter algebras and integro-differential algebras

Abstract

In a paper by Qi, Qin, Wang and Zhou, the authors studied a question which can be roughly stated as follows: Given an algebra A together with a Gröbner-Shirshov basis G, consider the free operated algebra B over A, such that the operator satisfies some polynomial identities Φ which are operated Gröbner-Shirshov in the sense of Guo et al., when does the union Φ ∪ G will be an operated Gröbner-Shirshov basis for B? The authors answered this question in the affirmative under a mild condition, and as a consequence they also got a linear basis of B.In this paper, we present a version of the above result for algebras endowed with several operators. As applications we get operated Gröbner-Shirshov bases for free differential Rota-Baxter algebras and free integro-differential algebras over algebras as well as their linear bases. One of key technical difficulties is to introduce new monomial orders for the case of two operators, which might be of independent interest. ZUAN LIU, ZIHAO QI, YUFEI QIN AND GUODONG ZHOU 4.4. Differential Rota-Baxter algebras vs integro-differential algebras 25 References 26