On the structure of cofree Hopf algebras

TL;DR

This paper extends classical theorems to non-cocommutative Hopf algebras, revealing their structure via B-infini-algebras and a universal enveloping functor, thus deepening understanding of cofree Hopf algebra construction.

Contribution

It introduces a new framework connecting cofree Hopf algebras with B-infini-algebras and constructs a universal enveloping functor U2 for these structures.

Findings

Cofree Hopf algebras are of the form U2(Prim H).

Primitive parts are B-infini-algebras.

Operad B-infini is described via planar trees.

Abstract

We prove an analogue of the Poincare'-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B-infini-algebra. We construct a universal enveloping functor U2 from B-infini-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the simple description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B-infini.