Construction of free differential algebras by extending Gr\"obner-Shirshov bases

TL;DR

This paper extends the construction of free differential algebras from sets to algebras by utilizing Gr"obner-Shirshov bases, enabling explicit basis construction for these more general free differential algebras.

Contribution

It demonstrates that Gr"obner-Shirshov basis properties of a base algebra can be extended to its free differential algebra, providing a new method for basis construction.

Findings

Extended Gr"obner-Shirshov basis property to free differential algebras

Established Poincaré-Birkhoff-Witt type basis for these algebras

Provided illustrative examples

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 generator-relation properties of a base algebra can be extended to the free differential algebra on this base algebra. More precisely, a Gr\"obner-Shirshov basis property of the base algebra can be extended to the free differential algebra on this base algebra, allowing a Poincar\'e-Birkhoff-Witt type basis for these more general free differential algebras. Examples are given as illustrations.