Bidendriform bialgebras, trees, and free quasi-symmetric functions

By Loïc Foissy. 2005

111 citations recorded by the source corpus.

Abstract 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).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2d542acf5aa6073de717cffe150894dffdec308a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

On quadri-algebras (2014, background): “By the bidendriform rigidity theorem [6, 7], (FQuad(V )∗,←,→) and (FQuad(V )∗, ↑ , ↓) are free dendriform algebras.”
Hopf algebras and dendriform structures arising from parking functions (2005, methodology): “These have been introduced by Foissy in [ 6 ].”