Coalgebras and bialgebras in combinatorics

Joni, S. A.; Rota, G.-C.

doi:10.1090/conm/006/646798

Cited by 181 publications

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“…The resulting Hopf algebra (F, ·, ∆, 1, ǫ, A) is a well known structure called the Faà di Bruno Hopf algebra [JR82] (see also [FGB05] in the context of renormalization).…”

Section: Hopf Algebra and Renormalizationmentioning

confidence: 99%

Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

Dütsch¹,

Fredenhagen²,

Keller³

et al. 2014

Journal of Mathematical Physics

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We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. This closed expression, which we call the Epstein-Glaser Forest Formula, is analogous to Zimmermann's Forest Formula for BPH renormalization. For scalar fields the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stückelberg and Petermann.

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“…The resulting Hopf algebra (F, ·, ∆, 1, ǫ, A) is a well known structure called the Faà di Bruno Hopf algebra [JR82] (see also [FGB05] in the context of renormalization).…”

Section: Hopf Algebra and Renormalizationmentioning

confidence: 99%

Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

Dütsch¹,

Fredenhagen²,

Keller³

et al. 2014

Journal of Mathematical Physics

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“…The definition of Newtonian coalgebra originated from Joni and Rota [13] under the name infinitesimal coalgebra. Our definition is from [7].…”

Section: Newtonian Coalgebrasmentioning

confidence: 99%

“…More importantly, P has a coalgebra structure. The pair formed by the star product * and the coproduct do not form a bialgebra, but instead a Newtonian coalgebra, a concept introduced by Joni and Rota [13]. The main observation we make is that the cd-index is a Newtonian coalgebra map from the vector space E spanned by all isomorphism classes of Eulerian posets to the algebra F of polynomials in the noncommutative variables c and d. We thus obtain that the prism operation corresponds to a certain derivation D on cd-polynomials, and the pyramid operation corresponds to a second derivation G. Hence, given the cd-index of a polytope, we may easily compute the cd-index of the prism and the pyramid of the polytope with the help of these two derivations.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Ehrenborg

Readdy

1998

Journal of Algebraic Combinatorics

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Abstract. The linear span of isomorphism classes of posets, P, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space P to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the ab-index of the Cartesian product of two posets given the ab-indexes of each poset.

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Section: Introductionmentioning

confidence: 99%

“…In its simplest form, the classical version gives the compositional inverse of an invertible formal power series. In other words, it expresses the antipode of the Hopf algebra of polynomial functions on the group of formal diffeomorphisms of the real line, also known as the Faá di Bruno algebra [5].Formal power series in one variable with coefficients in a noncommutative algebra can be composed (by substitution of the variable). This operation is not associative, so that they do not form a group.…”

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confidence: 99%

A One-Parameter Deformation of the Noncommutative Lagrange Inversion Formula

BULTEL

2011

Int. J. Algebra Comput.

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Abstract. We give a one-parameter deformation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-Frabetti-Krattenthaler for the antipode of the noncommutative Faá di Bruno algebra. Namely, we obtain a closed formula for the antipode of the one-parameter deformation of this Hopf algebra discovered by Foissy. IntroductionThe existence of combinatorial interpretations of the Lagrange inversion formula [12] can be traced back to the existence of noncommutative generalizations [4,11]. In its simplest form, the classical version gives the compositional inverse of an invertible formal power series. In other words, it expresses the antipode of the Hopf algebra of polynomial functions on the group of formal diffeomorphisms of the real line, also known as the Faá di Bruno algebra [5].Formal power series in one variable with coefficients in a noncommutative algebra can be composed (by substitution of the variable). This operation is not associative, so that they do not form a group. However, the analogue of the Faá di Bruno algebra still exists in this context. It is investigated in [1] in view of applications in quantum field theory. In [1], one finds in particular a combinatorial formula for its antipode. This formula is rederived by Novelli and Thibon [10], who also show that it is equivalent to the noncommutative Lagrange formula of Gessel and Pak-Postnikov-Retakh. They obtain it from the Brouder-Frabetti-Krattenthaler formula by a simple application of the antipode of the Hopf algebra of noncommutative symmetric functions.In [2], Foissy obtains, as a byproduct of his investigation of combinatorial SchwingerDyson equations, one-parameter families of Hopf algebras. They interpolate respectively between symmetric functions and Faá di Bruno, and between noncommutative symmetric functions and the noncommutative Faá di Bruno algebra.The main result of this paper is a closed formula for the antipode of the noncommutative family. As we shall see, this is a natural deformation of the Brouder-FrabettiKrattenthaler formula. As in the original version, we obtain a closed formula for the antipode of the simple complete noncommutative symmetric functions S n . Namely, we compute explicitely the coefficients α I (I n) in their expansion in the basis of complete noncommutative symmetric functions S I . As in the original version, we Date: May 17, 2011.

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Coalgebras and bialgebras in combinatorics

Cited by 181 publications

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Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

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A One-Parameter Deformation of the Noncommutative Lagrange Inversion Formula

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