No abstract

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ : H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H, ζ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized Dehn-Sommerville relations. We show that, for H = QSym, the generalized Dehn-Sommerville relations are the Bayer-Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the Malvenuto-Reutenauer Hopf algebra of permutations, the LodayRonco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of non-commutative symmetric functions.

An infinitesimal bialgebra is at the same time an associative algebra and coalgebra in such a way that the comultiplication is a derivation. This paper continues the basic study of these objects, with emphasis on the connections with the theory of Lie bialgebras. It is shown that non-degenerate antisymmetric solutions of the associative Yang᎐Baxter equation are in one to one correspondence with non-degenerate cyclic 2-cocycles. The associative and classical Yang᎐Baxter equations are compared: it is studied when a solution to the first is also a solution to the second. Necessary and sufficient conditions for obtaining a Lie bialgebra from an infinitesimal one are found, in terms of a canonical map that behaves simultaneously as a commutator and a cocommutator. The class of balanced infinitesimal bialgebras is introduced; they have an associated Lie bialgebra. Several well known Lie bialgebras are shown to arise in this way. The Ž construction of Drinfeld's double from earlier work by the author in press, in . Contemp. Math., Amer. Math. Soc., Providence for arbitrary infinitesimal bialgebras is complemented with the construction of the balanced double, for balanced ones. This construction commutes with the passage from balanced infinitesimal bialgebras to Lie bialgebras.

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. In this paper we define infinitesimal Hopf algebras, develop their basic theory and present several examples. It turns out that many properties of ordinary Hopf algebras possess an infinitesimal version. We introduce bicrossproducts, quasitriangular infinitesimal bialgebras, the corresponding infinitesimal Yang-Baxter equation and a notion of Drinfeld's double for infinitesimal Hopf algebras.

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasi-symmetric functions in terms of the facial structure of the cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed by permutations. Our results are obtained from a combinatorial description of the Hopf algebraic structure with respect to a new basis for this algebra, related to the original one via Möbius inversion on the weak order on the symmetric groups. This is in analogy with the relationship between the monomial and fundamental bases of the algebra of quasi-symmetric functions. Our results reveal a close relationship between the structure of the Malvenuto-Reutenauer Hopf algebra and the weak order on the symmetric groups.

We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the algebra of permutations, the shu e algebra, tensor products of dendriform algebras. We show that a pair of commuting Baxter operators on an associative algebra gives rise to a canonical quadri-algebra structure on the underlying space of the algebra. The main example is provided by the algebra End(A) of linear endomorphisms of an inÿnitesimal bialgebra A. This algebra carries a canonical pair of commuting Baxter operators: ÿ(T ) = T * id and (T ) = id * T , where * denotes the convolution of endomorphisms. It follows that End(A) is a quadri-algebra, whenever A is an inÿnitesimal bialgebra. We also discuss commutative quadri-algebras and state some conjectures on the free quadri-algebra.

23 pagesInternational audienceWe identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras

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 symmetric functions in the Malvenuto-Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra.We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopfalgebraic structure. In the dual basis for the graded dual Hopf algebra, our formula for the coproduct gives an explicit isomorphism with a free associative algebra. We also obtain a transparent proof of its isomorphism with the non-commutative Connes-Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes-Kreimer Hopf algebra is related to symmetric functions.