Some Hopf algebras of trees

Abstract

In the literature several Hopf algebras that can be described in terms of trees have been studied. This paper tries to answer the question whether one can understand some of these Hopf algebras in terms of a single mathematical construction. The starting point is the Hopf algebra of rooted trees as dened by Connes and Kreimer in [3] (section 2). Apart from its physical relevance, it has a universal property in Hochschild cohomology. We generalize the operadic construction by Moerdijk [12] of this Hopf algebra to more general trees (with colored edges), and prove a universal property in coalgebra Hochschild cohomology (sections 3, 4, and 5). For a Hopf operad P , the construction is based on the operad P [n] obtained from P by adjoining a free n-ary operation. The dierence with respect to [12] is that we also prove that the operad P [n] is under suitable restrictions a Hopf operad. This presentation simplies some proofs and assures that relevant unctorial properties come for free. Section 6 shows that we can do with less restrictions if we are only interested in the the initial algebra P [n](0).