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28
Combinatorial Hopf algebras and generalized DehnSommerville relations
, 2003
"... 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 u ..."
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Cited by 95 (20 self)
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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 quasisymmetric 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 DehnSommerville relations. We show that, for H = QSym, the generalized DehnSommerville relations are the BayerBillera 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 MalvenutoReutenauer Hopf algebra of permutations, the
On Posets and Hopf Algebras
 Adv. Math
, 1996
"... 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 ..."
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Cited by 81 (13 self)
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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
RotaBaxter algebras in renormalization of perturbative quantum field theory
 Fields Institute Communications v. 50, AMS
"... 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 regularize ..."
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Cited by 35 (11 self)
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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 multiplezetavalues and matrix differential equations.
Noncommutative Pieri Operators On Posets
 J. Combin. Th. Ser. A
, 2000
"... We consider graded representations of the algebra NC of noncommutative symmetric functions on the Zlinear 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 quasisymmetric fun ..."
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Cited by 29 (5 self)
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We consider graded representations of the algebra NC of noncommutative symmetric functions on the Zlinear 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 quasisymmetric functions.
Quantum groups and quantum field theory: I. The free . . .
, 2002
"... 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 twopoint function and the Feyn ..."
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Cited by 22 (0 self)
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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 twopoint function and the Feynman propagator of scalar fields to reproduce the operator product and the timeordered 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.
New Results for the Martin Polynomial
, 1998
"... 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 graph ..."
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Cited by 21 (9 self)
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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 skeintype 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 Hopfalgebraic 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.
A treatise on quantum Clifford Algebras
, 2002
"... ... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are ve ..."
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Cited by 19 (14 self)
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... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A ‘spinorial ’ generalization of the antipode is given. The nonexistence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially generated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordization is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory.
Combinatorial Hopf algebras
"... Abstract. We give a precise definition of “combinatorial Hopf algebras”, and we classify them in the four cases: associative or commutative, general or rightsided. For instance a cofreecocommutative combinatorial Hopf algebra is completely determined by its primitive part which is a preLie algebr ..."
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Cited by 14 (2 self)
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Abstract. We give a precise definition of “combinatorial Hopf algebras”, and we classify them in the four cases: associative or commutative, general or rightsided. For instance a cofreecocommutative combinatorial Hopf algebra is completely determined by its primitive part which is a preLie algebra. The classification gives rise to several good triples of operads. It involves the operads: dendriform, preLie, brace, GerstenhaberVoronov, and variations of them.
Hopf Algebras of Combinatorial Structures
 Canadian Journal of Mathematics
, 1993
"... 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 s ..."
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Cited by 12 (1 self)
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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.
Combinatorial Models For Coalgebraic Structures
 Adv. Math
, 1997
"... . We introduce a convenient category of combinatorial objects, known as cellsets, 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 ..."
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Cited by 10 (5 self)
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. We introduce a convenient category of combinatorial objects, known as cellsets, 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 CWcomplex, 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...