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

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.