Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated with a pre-Lie structure on formal power series. We study thes...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2023 |
| ISSN: | 1815-0659 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211905 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability. Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia and Lorenzo Zambotti. SIGMA 19 (2023), 038, 17 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862740307903250432 |
|---|---|
| author | Ebrahimi-Fard, Kurusch Patras, Frédéric Tapia, Nikolas Zambotti, Lorenzo |
| author_facet | Ebrahimi-Fard, Kurusch Patras, Frédéric Tapia, Nikolas Zambotti, Lorenzo |
| citation_txt | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability. Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia and Lorenzo Zambotti. SIGMA 19 (2023), 038, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated with a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.
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| first_indexed | 2026-04-17T17:39:49Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211905 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T17:39:49Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ebrahimi-Fard, Kurusch Patras, Frédéric Tapia, Nikolas Zambotti, Lorenzo 2026-01-16T10:54:41Z 2023 Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability. Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia and Lorenzo Zambotti. SIGMA 19 (2023), 038, 17 pages 1815-0659 2020 Mathematics Subject Classification: 16T05; 16T10; 16T30; 17A30; 46L53; 46L54 arXiv:2204.01445 https://nasplib.isofts.kiev.ua/handle/123456789/211905 https://doi.org/10.3842/SIGMA.2023.038 We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated with a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability. This work was partially supported by the project “Pure Mathematics in Norway”, funded by the Trond Mohn Foundation and the Tromsø Research Foundation. KEF was supported by the Research Council of Norway through the project 302831 “Computational Dynamics and Stochastics on Manifolds” (CODYSMA). NT was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy– The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). KEF and NT would also like to thank the Department of Mathematics at the Saarland University for warm hospitality during a sabbatical visit. FP acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Duall project, grant agreement No. 670624), from the ANR project Algebraic Combinatorics, Renormalization, Free probability and Operads– CARPLO (Project-ANR-20-CE40-0007), and from the ANR– FWF project PAGCAP. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability Article published earlier |
| spellingShingle | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability Ebrahimi-Fard, Kurusch Patras, Frédéric Tapia, Nikolas Zambotti, Lorenzo |
| title | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability |
| title_full | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability |
| title_fullStr | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability |
| title_full_unstemmed | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability |
| title_short | Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability |
| title_sort | shifted substitution in non-commutative multivariate power series with a view toward free probability |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211905 |
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