Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics
We develop an algebraic theory of colored, semigroup-like-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 cond...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211734 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics. Ralph M. Kaufmann and Yang Mo. SIGMA 18 (2022), 053, 42 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862741838424702976 |
|---|---|
| author | Kaufmann, Ralph M. Mo, Yang |
| author_facet | Kaufmann, Ralph M. Mo, Yang |
| citation_txt | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics. Ralph M. Kaufmann and Yang Mo. SIGMA 18 (2022), 053, 42 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We develop an algebraic theory of colored, semigroup-like-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 are needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. To construct antipodes, we discuss formal localization constructions and quantum deformations. These allow us 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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| first_indexed | 2026-04-17T18:04:09Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211734 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T18:04:09Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kaufmann, Ralph M. Mo, Yang 2026-01-09T12:55:13Z 2022 Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics. Ralph M. Kaufmann and Yang Mo. SIGMA 18 (2022), 053, 42 pages 1815-0659 2020 Mathematics Subject Classification: 16T05; 18M85; 81T15; 81R50 arXiv:2104.08895 https://nasplib.isofts.kiev.ua/handle/123456789/211734 https://doi.org/10.3842/SIGMA.2022.053 We develop an algebraic theory of colored, semigroup-like-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 are needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. To construct antipodes, we discuss formal localization constructions and quantum deformations. These allow us 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. This paper is the outgrowth of a long lasting conversations with Dirk Kreimer, whom we would like to congratulate and profoundly thank for his interest and support. Much of the impetus for this work has come through interactions with him and his group, and it would not have come into existence without Dirk. RK wishes to thank the Humboldt University, the KMPB, and the Humboldt Foundation for making these visits possible. We furthermore would like to thank F. Brown, K. Yeats, and C. Berger for helpful discussions and the organizers of the conference Algebraic Structures in Perturbative Quantum Field Theory at the IHES, in November 2020. The wonderful interactions that took place there led to some of the results of this paper. We also wish to thank the referees for their input, which was very beneficial for the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics Article published earlier |
| spellingShingle | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics Kaufmann, Ralph M. Mo, Yang |
| title | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics |
| title_full | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics |
| title_fullStr | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics |
| title_full_unstemmed | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics |
| title_short | Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics |
| title_sort | pathlike co/bialgebras and their antipodes with applications to bi- and hopf algebras appearing in topology, number theory and physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211734 |
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