Composition-Diamond lemma for $\lambda$-differential associative algebras with multiple operators

TL;DR

This paper develops a Composition-Diamond lemma for $$-differential associative algebras with multiple operators, enabling the construction of free $$-differential Rota-Baxter algebras and their bases.

Contribution

It introduces a new lemma for $$-differential associative algebras with multiple operators, facilitating the construction of free algebras with explicit bases.

Findings

Established the Composition-Diamond lemma for $$-differential associative algebras

Derived Grb6bner-Shirshov bases for free $$-differential Rota-Baxter algebras

Constructed linear bases and free algebras using words

Abstract

In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential Rota-Baxter algebras. In particular, linear bases of free $\lambda$-differential Rota-Baxter algebras are obtained and consequently, the free $\lambda$-differential Rota-Baxter algebras are constructed by words.