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 quasi-symmetric 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

this paper have a finite number of elements, a minimum element 0 # and a maximum element 1 . For two elements x and y in a poset P, such that x#y, define the interval [x, y]=[z#P:x#z#y]. We will consider [x, y] as a poset, which inherits its order relation from P. Observe that [x, y] is a poset with minimum element x and maximum element y. For x, y # P such that x#y, we may define the Mo# bius function +(x, y) recursively by +(x, y)= & : x#z<y +(x, z), if x<y, 1, if x=y

Algebraic techniques are used to find several new combinatorial interpretations for valuations of the Martin polynomial, M(G; s), for unoriented graphs. The Martin polynomial of a graph, introduced by Martin in his 1977 thesis, encodes information about the families of closed paths in Eulerian graphs. The new results here are found by showing that the Martin polynomial is a translation of a universal skein-type graph polynomial P(G) which is a Hopf map, and then using the recursion and induction which naturally arise from the Hopf algebra structure to extend known properties. Specifically, when P(G) is evaluated by substituting s for all cycles and 0 for all tails, then P(G) equals sM(G; s+2) for all Eulerian graphs G. The Hopf-algebraic properties of P(G) are then used to extract new properties of the Martin polynomial, including an immediate proof for the formula for M(G; s) on disjoint unions of graphs, combinatorial interpretations for M(G; 2+2 k) and M(G; 2&2 k) with k # Z 0, and a new formula for the number of Eulerian orientations of a graph in terms of the vertex degrees of its Eulerian subgraphs.

Abstract. Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. In this context the notion of Rota–Baxter algebras enters the scene. We review several aspects of Rota–Baxter algebras as they appear in other sectors also relevant to perturbative renormalization, for instance multiple-zeta-values and matrix differential equations.

We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra generated by the intervals of P to the Hopf algebra of quasi-symmetric functions.

Abstract. The quantum field algebra of real scalar fields is shown to be an example of infinite dimensional quantum group. The underlying Hopf algebra is the symmetric algebra S(V) and the product is Wick’s normal product. Two coquasitriangular structures can be built from the two-point function and the Feynman propagator of scalar fields to reproduce the operator product and the time-ordered product as twist deformations of the normal product. A correspondence is established between the quantum group and the quantum field concepts. On the mathematical side the underlying structures come out of Hopf algebra cohomology.

We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper

. We introduce a convenient category of combinatorial objects, known as cell-sets, on which we study the properties of the appropriate free abelian group functor. We obtain a versatile generalization of the notion of incidence coalgebra, giving rise to an abundance of coalgebras, Hopf algebras, and comodules, all of whose structure constants are positive integers with respect to certain preferred bases. Our category unifies and extends existing constructions in algebraic combinatorics, providing proper functorial descriptions; it is inspired in part by the notion of CW-complex, and is also geared to future applications in algebraic topology and the theory of formal group laws. 1. Introduction The theory of coalgebras and Hopf algebras was first developed by algebraic topologists more than fifty years ago. Since the seminal work of Joni and Rota [13], applications to combinatorial mathematics have grown steadily in prominence, motivated by the principle that a diagonal, or coproduct ma...

Abstract. The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B) +), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B) +), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S 1 (B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B) +) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B) +) is shown to give the same results as the standard renormalization procedure for the scalar field. 1.

Abstract: A generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as morphisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.