Multiple polylogarithms, polygons, trees and algebraic cycles

Gangl, Herbert; Goncharov, A. B.; Levin, A.

doi:10.1090/pspum/080.2/2483947

Cited by 18 publications

(72 citation statements)

References 14 publications

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“…Here we review this operation and show how it can be graphically interpreted. The graphical notation we present in turn connects to the graphical notation in [Gon05] and [GGL09]. The first of the two subcategories of ∆ + is ∆ and the second is the category of intervals.…”

Section: Appendix B Coalgebras and Hopf Algebrasmentioning

confidence: 99%

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Gálvez-Carrillo

Kaufmann

Tonks

2020

Communications in Number Theory and Physics

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We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

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Section: Appendix B Coalgebras and Hopf Algebrasmentioning

confidence: 99%

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Gálvez-Carrillo

Kaufmann

Tonks

2020

Communications in Number Theory and Physics

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“…, z n ) in a similar fashion, but we need to introduce trees with two different kinds of external edges, and an accordingly modified forest cycling map gives us the associated algebraic cycles (cf. [3]). …”

Section: Discussionmentioning

confidence: 99%

“…For a proof, we refer to [3]; the main idea is that at each internal vertex we have at least one incoming and one outgoing edge, which implies that their respective coordinates in the associated cycle cover up for each other.…”

Section: Definition 43mentioning

confidence: 99%

“…This property ensures, cf. [3], that one can build from τ (x 1 , ..., x m ) an element in the Hopf algebra χ mot of [1]. Here is how the cancellations take place for m = 3.…”

Section: Example 52mentioning

confidence: 99%

“…

This is a short exposition-mostly by way of the toy models "double logarithm" and "triple logarithm"-which should serve as an introduction to the article [3] in which we establish a connection between multiple polylogarithms, rooted trees and algebraic cycles.

…”

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confidence: 99%

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Multiple Logarithms, Algebraic Cycles and Trees

Gangl

Goncharov

Levin³

Frontiers in Number Theory, Physics, and Geometry II

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Dichotomy of the Addition of Natural Numbers

Loday

2012

Progress in Mathematics

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This is an elementary presentation of the arithmetic of trees. We show how it is related to the Tamari poset. In the last part we investigate various ways of realizing this poset as a polytope (associahedron), including one inferred from Tamari's thesis.This construction gives a section to ψ that we denote byProposition 3. The Tamari polytope KT n is a hypercube shaped polytope, with extremal points M(σ (α)), for α ∈ {±} n . For any tree t the point M(t) lies on a face of this hypercube containing M(σ ψ(t)).Proof. It is easily seen by induction that the convex hull of the points M(σ (α)) form a (combinatorial) hypercube.Up to a change of orientation, this is the cubical version of the associahedron described in [17] section 2.5 (see also [19] Appendix 1). It is also described in [27].

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Multiple polylogarithms, polygons, trees and algebraic cycles

Abstract: We construct, for a field

Cited by 18 publications

References 14 publications

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Multiple Logarithms, Algebraic Cycles and Trees

Dichotomy of the Addition of Natural Numbers

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