A Milnor–Moore theorem for dendriform Hopf algebras

Abstract

A dendriform Hopf algebra is a Hopf algebra, such that the product ∗ is the sum of two operations ≺ and ≻, verifying certain conditions between them and with the coproduct Δ. The purpose of this Note is to announce a Milnor–Moore style theorem for these algebras. The role of Lie algebras is played by brace algebras, which are defined by n-ary operations (one for each n≥2) satisfying some relations. We show that a dendriform Hopf algebra is isomorphic to the envelopping algebra of its brace algebra of primitive elements. One of the ingredients of the proof is the construction of Eulerian idempotents in this context.