Differential Algebra Structures on Familes of Trees

Abstract

It is known that the vector space spanned by labeled rooted trees forms a Hopf algebra. Let k be a field and let R be a commutative k-algebra. Let H denote the Hopf algebra of rooted trees labeled using derivations D in Der(R). In this paper, we introduce a construction which gives R a H-module algebra structure and show this induces a differential algebra structure of H acting on R. The work here extends the notion of a R/k-bialgebra introduced by Nichols and Weisfeiler.