Hopf algebras, from basics to applications to renormalization

Dominique Manchon

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Hopf algebras, from basics to applications to renormalization

Dominique Manchon

Quantum Algebra · Submitted 30 Aug 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It consists in a presentation of some aspects of Connes' and Kreimer's work on renormalization in the context of general connected Hopf algebras, in particular Birkhoff decomposition and, in the graded case, the scattering-type formula.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

An extended version of a series of lectures given at Bogota in december 2002. It consists in a presentation of some aspects of Connes' and Kreimer's work on renormalization in the context of general connected Hopf algebras, in particular Birkhoff decomposition and, in the graded case, the scattering-type formula.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An extended version of a series of lectures given at Bogota in december 2002.	background 0.99	0.39	0.02	asserted 0.94
s1	It consists in a presentation of some aspects of Connes' and Kreimer's work on renormalization in the context of general connected Hopf algebras, in particular Birkhoff decomposition and, in the graded case, the scattering-type formula.	contribution 0.90	0.17	0.26	asserted 0.99

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The paper

Original abstract

An extended version of a series of lectures given at Bogota in december 2002. It consists in a presentation of some aspects of Connes' and Kreimer's work on renormalization in the context of general connected Hopf algebras, in particular Birkhoff decomposition and, in the graded case, the scattering-type formula.

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References

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams
math.CO 2020-09 conditional novelty 8.0 of 10

The number of 2-connected chord diagrams on n chords is asymptotically e^{-2}(2n-1)!!(1 - 6/(2n-1) - 4/((2n-3)(2n-1)) - ...), derived from a new generating-function recurrence.
Feynman integrals and hyperlogarithms
math-ph 2015-06 accept novelty 8.0 of 10

Massless ladder-box and vertex-width-three Feynman integrals are proven to reduce to multiple polylogarithms to all orders in the epsilon expansion, with an implemented hyperlogarithm integration method.
Piecewise Symmetric Tensors
math.RT 2026-07 accept novelty 7.0 of 10

Every k-tensor is the unique sum of an m-piecewise-symmetric tensor and a (k-m)-piecewise-alternating tensor; the latter space is the annihilator of level-k signatures of m-segment paths.
Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groups
cs.LG 2025-09 unverdicted novelty 7.0 of 10

Introduces the first explicit near-reversible integrator for neural SDEs on Lie groups by extending EES schemes with Bazavov's commutator-free lift, achieving better stability and up to 10x memory reduction on manifol...
Hopf Algebras in Combinatorics
math.CO 2014-09 conditional novelty 3.0 of 10

A graduate survey organizes the main combinatorial Hopf algebras and their standard theorems into a single course text.

Show 1 more citing papers

Feynman Integrals
hep-th 2022-01 accept novelty 2.0 of 10

A self-contained graduate course assembling the modern methods for computing Feynman integrals, from graph polynomials and differential equations through symbols, cluster algebras, elliptic curves and motives.

How the literature uses it

25 arXiv papers cite this in 84 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 49 sentences without a confident label

Uses its method or tool 2 papers

Here we use the second construction following [ref] where one keeps all the residues $r∈^*$ which behave as group-like elements w.r.t. the coproduct [ref] , $r=r⊗r$, and augment $$ by formal inverses $r^-1$ for all $r∈^*$ defining $r^-1 r = r r^-1=$.

Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs · The Hopf algebra of 2-graphs and renormalization in cNLFT · 0.95
The antipode strategy for this job amounts to looking at the character $χ_mc-bc ∈( )$ which corresponds to $$, and then by performing the required inversion via the formula (cf. [ref] ) [equation]

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants · A discussion of the antipode of $$ · 0.97

Cites it as background 18 papers

We will follow mainly [ref] (see also [ref] ) for Hopf algebras of forests and [ref] for the more general theory of Hopf algebras; when details and proofs are omitted it is intended that they are to be found therein.

A Transfer Principle for Branched Rough Paths · The Connes-Kreimer and Grossman-Larson Hopf algebras · 0.88
In the first section we will merely state the basic definitions and properties of bialgebras and refer to [ref] for details and omitted proofs.

Hopf algebraic Renormalization of Kreimer's toy model · Introduction · 1.00
[ref] ) A bialgebra $(H,,)$ with unit map $$ and counit $$ together with an antipode, a linear operator $: H→H$[S]$$Antipode of an Hopf algebra fulfilling the antipode propertyantipode!property [equation] is called a Hopf algebra.

From Hopf algebras to rough paths and regularity structures · Bialgebras and Hopf algebras · 0.80

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