Christian Brouder, Alessandra Frabetti
In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semi-direct coproduct of Hopf algebras.
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We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward-Takahashi identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT-identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT-identities, which leads in particular to the well-known identity $Z_1=Z_2$.
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.
We study the renormalization of massless QED from the point of view of the Hopf algebra discovered by D. Kreimer. For QED, we describe a Hopf algebra of renormalization which is neither commutative nor cocommutative. We obtain explicit renormalization formulas for the electron and photon propagators, for the vacuum polarization and the electron self-energy, which are equivalent to Zimmermann's forest formula for the sum of all Feynman diagrams at a given order of interaction. Then we extend to QED the Connes-Kreimer map defined by the coupling constant of the theory (i.e. the homomorphism betw…
We produce an explicit embedding of the planar algebra of the Drinfeld double of a finite-dimensional, semisimple and cosemisimple Hopf algebra $H$ into the two-cabling of the planar algebra of the dual Hopf algebra $H^*$ and characterise the image.
This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra …
We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates duality in planar graphs and Hopf algebras, is the subject of a final section.
In \cite{Kreimer1,Connes,Broadhurst,Kreimer2}, a commutative, non cocommutative Hopf algebra H_R of (decorated) rooted trees was introduced. It is related to the Hopf algebra H_CM introduced in \cite{Moscovici}. Its dual Hopf algebra is the enveloping algebra of the Lie algebra of rooted trees L^1. In this paper, we introduce a non commutative, non cocommutative Hopf algebra H_PR of decorated planar rooted trees. We first show that H_PR is self dual. We then use this result to construct a non degenerate bilinear form (,) on H_PR, which respects the Hopf algebra structure of H_PR. Moreover, we …
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 t…
This paper generalizes the operadic construction of the Connes-Kreimer Hopf algebra of rooted trees by Moerdijk. Examples of Hopf algebras obtained in this way include the Loday-Ronco Hopf algebra of planar binary trees and the Brouder-Frabetti pruning Hopf algebra. In some examples we obtain a natural pre-Lie or dendriform algebra structure on the dual Hopf algebra. v2: Proof in section 8 corrected. v3: more corrections.
Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.
We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral…
We prove an analogue of the Poincare'-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B-infini-algebra. We construct a universal enveloping functor U2 from B-infini-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the simple description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B-infini.
In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and s…
We review the relation between the Hopf algebra QSym of quasi-symmetric functions and the multiple zeta values, and then discuss a commutative diagram involving the Hopf algebra Sym of symmetric functions, the Hopf algebra dual NSym of QSym, and the Hopf algebras of rooted trees and planar rooted trees as defined by Kreimer and Foissy respectively.
In "Hopf algebra of the planar binary trees", Adv. Math. 139 (1998), no. 2, 293--309, we constructed by induction a graded associative product on the vector space generated by the planar binary trees (resp. the permutations). In the present paper we prove an explicit formula for this product in terms of a partial order on the set of planar binary trees (resp. the weak Bruhat order on the symmetric groups).
We study actions of ``compact quantum groups'' on ``finite quantum spaces''. According to Woronowicz and to general $\c^*$-algebra philosophy these correspond to certain coactions $v:A\to A\otimes H$. Here $A$ is a finite dimensional $\c^*$-algebra, and $H$ is a certain special type of Hopf *-algebra. If $v$ preserves a positive linear form $ϕ:A\to\c$, a version of Jones' ``basic construction'' applies. This produces a certain $\c^*$-algebra structure on $A^{\otimes n}$, plus a coaction $v_n :A^{\otimes n}\to A^{\otimes n}\otimes H$, for every $n$. The elements $x$ satisfying $v_n(x)=x\otimes …
This paper is a brief mathematical excursion which starts from quantum electrodynamics and leads to the Moebius function of the Tamari lattice of planar binary trees, within the framework of groups of tree-expanded series. First we recall Brouder's expansion of the photon and the electron Green's functions on planar binary trees, before and after the renormalization. Then we recall the structure of Connes and Kreimer's Hopf algebra of renormalization in the context of planar binary trees, and of their dual group of tree-expanded series. Finally we show that the Moebius function of the Tamari p…
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $F$ equipped with a character (multiplicative linear functional) $ζ:H\to F$. 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 …
A Quantization of the Loday-Ronco Hopf Algebra
本文基于 Eynard-Orantin 拓扑递归构造 Loday-Ronco Hopf 代数的量子化 $k[Y^\infty]_h$，证明其 Hopf 代数性质并刻画拓扑递归解空间的商子代数结构。
The N-variable Hopf algebra introduced by Brouder, Fabretti, and Krattenaler (BFK) in the context of non-commutative Lagrange inversion can be identified with the inverse of the incidence algebra of N-colored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with non-commuting coefficients and N non-commuting variables. As in the case of the Faa di Bruno Hopf algebra, there is an analogue of the Zimmermann cancellation formula. The summands of the (BFK) antipode can indexed by the depth first ordering of vertic…
The aim of this paper is to further explore an idea from J.-L. Loday briefly exposed in [5]. We impose a natural and simple symmetry on a unit action over the most general quadratic relation which can be written. This leads us to two families of binary, quadratic and regular operads whose free objects, as well as their duals in the sense of Ginzburg and Kapranov are computed. Roughly speaking, free objects found here are in relation to $m$-ary trees, triangular numbers and more generally $m$-tetrahedral numbers, homogeneous polynomials on $m$ commutative indeterminates over a field $K$ and pol…
The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the multiplication, while the second group is the set of formal diffeomorphisms with the composition. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra gi…
Old and New Minimalism: a Hopf algebra comparison
本文通过Hopf代数比较极简主义的旧新表述，证明新表述中内外部Merge源自同一运算，结构更优，并对比了外化模型与平面嵌入方案。
We describe quasi-Hopf twist deformations of flat closed string compactifications with non-geometric R-flux using a suitable cochain twist, and construct nonassociative deformations of fields and differential calculus. We report on our new findings in using this formalism to construct perturbative nonassociative field theories on these backgrounds. We describe the modifications to the usual classification of Feynman diagrams into planar and non-planar graphs. The example of phi(4) theory is studied in detail and the one-loop contributions to the two-point function are calculated.
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