Construction of free differential algebras by extending Gröbner-Shirshov bases. (November 2021)

• Record Type: Journal Article
• Authors: Li, Yunnan; Guo, Li
• Is Part Of: Journal of symbolic computation. Volume 107(2021)
• Page Start: 167
• Page End: 189
• Publication Date: 2021-11
• Subjects: 13Nxx -- 12H05 -- 16S15 -- 16S10 -- 16W99; Differential algebra -- Free differential algebra -- Gröbner basis -- Gröbner-Shirshov basis; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.03.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation presentation of a base algebra can be extended to the free differential algebra on this base algebra. More precisely, a Gröbner-Shirshov basis property of the base algebra can be extended to the free differential algebra on this base algebra, allowing a Poincaré-Birkhoff-Witt type basis for these more general free differential algebras. Examples are provided as illustrations.