Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

Abstract

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma.

Communicated by P. Kolesnikov

KEYWORDS:

• Differential algebra
• differential Rota-Baxter algebra
• Gröbner-Shirshov basis
• quasi-idempotent operator
• Rota-Baxter algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W99
• 16S10
• 13P10
• 16Z10
• 12H05

Acknowledgments

We thank the referee for very valuable suggestions.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11961031 and 12326324) and Jiangxi Provincial Natural Science Foundation (Grant No. 20224BAB201003).