Bidendriform bialgebras, trees, and free quasi-symmetric functions

Loïc Foissy (1)

1 LMR - Laboratoire de Mathématiques de Reims

We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).

Mots clés: dendriform algebras; trees; Malvenuto-Reutenauer Hopf algebra

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. Bidendriform bialgebras, trees, and free quasi-symmetric functions. Journal of Pure and Applied Algebra, 2007. ⟨hal-00004902⟩