Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

Ralph M. Kaufmann; Yang Mo

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Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

Ralph M. Kaufmann, Yang Mo

Quantum Algebra · Submitted 18 Apr 2021

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

The authors' main claim This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics.

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arXiv ID2104.08895 (version 3)

Pith IDMVW3NUNN, Pith's permanent record of this paper and its review

CitationRalph M. Kaufmann, Yang Mo. "Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics." arXiv:2104.08895 (2021). https://pith.science/paper/MVW3NUNN

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What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.19 · mentions code or data availability 0.02.

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We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras.	contribution 1.00	0.95	0.21	asserted 1.00
s1	This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics.	interpretation 0.79	0.30	0.20	asserted 1.00
s2	In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer.	result 0.53	0.89	0.21	asserted 0.92
s3	These are met in the relevant examples.	result 0.96	0.59	0.09	asserted 1.00
s4	In order to construct antipodes, we discuss formal localization constructions and quantum deformations.	method 0.92	0.82	0.05	asserted 0.98
s5	These allow to define and explain the appearance of Brown style coactions.	interpretation 0.38	0.77	0.06	asserted 0.97
s6	Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.	contribution 0.56	0.89	0.23	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 0.99.

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Original abstract

We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.

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2 arXiv papers cite this in 2 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

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Cites it as background 1 paper

Here colored means that there are source and target maps and they have to coincide, cf. e.g. [ref] .

Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories · Valid formulas · 0.81

What it cites · 69 sentences without a confident label

Builds on 1 paper

This was generalized in [ref] for $B$ any connected bialgebra and a character that is the exponential of an infinitesimal character.

Integrable Renormalization II: the general case · Recollections on renormalization via characters · 0.82

Cites as support 5 papers

These are also motivated by the natural examples [ref] .

Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem · Quotients and quantum deformations for grouplike central and invertible characters · 0.96
These are also motivated by the natural examples [ref] .

Hopf algebra approach to Feynman diagram calculations · Quotients and quantum deformations for grouplike central and invertible characters · 0.96
These are also motivated by the natural examples [ref] .

Hopf Algebras, Renormalization and Noncommutative Geometry · Quotients and quantum deformations for grouplike central and invertible characters · 0.96

Cites as background 14 papers

These coalgebras were analyzed in [ref] , and their history goes back to [ref] , see also [ref] .

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects · Categorical coalgebra · 1.00
These coalgebras were analyzed in [ref] , and their history goes back to [ref] , see also [ref] .

Feynman Categories · Categorical coalgebra · 1.00
A renormalization scheme in the Connes–Kreimer formalization of the BPHZ renormalization [ref] , is defined on the convolution algebra of a bialgebra with a Rota–Baxter (RB) algebra, see also [ref] .

A combinatorial Hopf algebra for nonlinear output feedback control systems · Recollections on renormalization via characters · 0.96

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