Groebner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras

TL;DR

This paper develops a Composition-Diamond lemma for associative algebras with multiple operators and constructs Groebner-Shirshov bases for free Rota-Baxter, differential, and combined algebras, providing explicit linear bases.

Contribution

It introduces a new Composition-Diamond lemma for multi-operator associative algebras and derives Groebner-Shirshov bases for several important free algebras, connecting to recent results.

Findings

Established the Composition-Diamond lemma for associative algebras with multiple operators

Obtained Groebner-Shirshov bases for free Rota-Baxter, differential, and combined algebras

Explicit linear bases for these free algebras are constructed

Abstract

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $\lambda$-differential algebra and $\lambda$-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to those obtained by Ebrahimi-Fard and Guo, and Guo and Keigher recently by using other methods.