Eulerian idempotents and Milnor–Moore theorem for certain non-cocommutative Hopf algebras

The purpose of this paper is to prove a Milnor–Moore style theorem for a particular kind of non-cocommutative Hopf algebras: the dendriform algebras. 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 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 enveloping algebra of its brace algebra of primitive elements. One of the ingredients of the proof is the construction of Eulerian idempotents in this context.