We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitiveâ€¦ (More)

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) Î¶ : H â†’ k. We show that the terminal object in the categoryâ€¦ (More)

We show the existence of a unital subalgebra Pn of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that Pn is theâ€¦ (More)

Infinitesimal bialgebras were introduced by Joni and Rota [J-R]. An infinitesimal bialgebra is at the same time an algebra and a coalgebra, in such a way that the comultiplication is a derivation. Inâ€¦ (More)

Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetricâ€¦ (More)

We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form aâ€¦ (More)

Joyals notion of species constitutes a good framework for the study of certain algebraic structures associated to combinatorial objects. We discuss the notion of Hopf monoid in the category ofâ€¦ (More)

This study quantitatively evaluated the fluorescence intensity of resin composites with different opacities and translucencies and determined changes in fluorescence after accelerated aging, usingâ€¦ (More)

We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of theâ€¦ (More)