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Symmetry, Integrability and Geometry: Methods and Applications, 2023, Volume 19, 038, 17 pp.
DOI: https://doi.org/10.3842/SIGMA.2023.038 (Mi sigma1933) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

Kurusch Ebrahimi-Farda, Frédéric Patrasb, Nikolas Tapiacd, Lorenzo Zambottie

a Department of Mathematical Sciences, Norwegian University of Science and Technology, NO 7491 Trondheim, Norway
b Université Côte d'Azur, CNRS, UMR 7351, Parc Valrose, 06108 Nice Cedex 02, France
c Weierstraß-Institut Berlin, Berlin, Germany
d Technische Universität Berlin, Berlin, Germany
e LPSM, Sorbonne Université, CNRS, Université Paris Cité, 4 Place Jussieu, 75005 Paris, France
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DOI: https://doi.org/10.3842/SIGMA.2023.038
Abstract: 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 to 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.
Keywords: non-commutative probability theory, non-commutative power series, moments and cumulants, combinatorial Hopf algebra, pre-Lie algebra.
Funding agency Grant number
Trond Mohn Foundation
Tromsø Research Foundation
Research Council of Norway 302831
Deutsche Forschungsgemeinschaft 390685689
European Research Council 670624
Agence Nationale de la Recherche ANR-20-CE40-0007
PAGCAP
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 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). 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.
Received: April 5, 2022; in final form May 29, 2023; Published online June 8, 2023
Document Type: Article
MSC: 16T05, 16T10, 16T30, 17A30, 46L53, 46L54
Language: English
Citation: Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia, Lorenzo Zambotti, “Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability”, SIGMA, 19 (2023), 038, 17 pp.
Citation in format AMSBIB
\Bibitem{EbrPatTap23}
\by Kurusch~Ebrahimi-Fard, Fr\'ed\'eric~Patras, Nikolas~Tapia, Lorenzo~Zambotti
\paper Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
\jour SIGMA
\yr 2023
\vol 19
\papernumber 038
\totalpages 17
\mathnet{http://mi.mathnet.ru/sigma1933}
\crossref{https://doi.org/10.3842/SIGMA.2023.038}
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