Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-23T18:06:45.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Wick polynomials in noncommutative probability: a group-theoretical approach

Part of: General nonassociative rings Selfadjoint operator algebras Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  25 August 2021

Kurusch Ebrahimi-Fard ,
Frédéric Patras ,
Nikolas Tapia  and
Lorenzo Zambotti
Kurusch Ebrahimi-Fard
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway e-mail: kurusch.ebrahimi-fard@ntnu.no
Frédéric Patras
Affiliation:
Laboratoire J.A. Dieudonné, Université Côte d’Azur, CNR, UMR 7351, Parc Valrose, Nice, France e-mail: frederic.patras@unice.fr
Nikolas Tapia*
Affiliation:
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany and Institute of Mathematics, Technische Universtät Berlin, Straße des 17. Juni 136, 10587 Berlin, Germany
Lorenzo Zambotti
Affiliation:
Laboratoire de Probabilités, Statistiques et Modélisation, Sorbonne Université, Université de Paris, 4 Place Jussieu, 75005 Paris, France e-mail: zambotti@lpsm.paris
*
e-mail: tapia@wias-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

Keywords

Wick polynomialsmonotone cumulantsfree cumulantsBoolean cumulantsformal power seriescombinatorial Hopf algebrashuffle algebragroup actions

MSC classification

Primary: 17A30: Algebras satisfying other identities 46L53: Noncommutative probability and statistics 46L54: Free probability and free operator algebras
Secondary: 16T05: Hopf algebras and their applications 16T10: Bialgebras 16T30: Connections with combinatorics

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 6 , December 2022 , pp. 1673 - 1699
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work was supported by the European Research Council for Informatics and Mathematics through contract ERCIM 2018-10, and the BMS MATH+ EF1-5 project “On robustness of Deep Neural Networks.”

References

Akhiezer, N. I., The classical moment problem: and some related questions in analysis, University Mathematical Monographs, 5, Oliver and Boyd, Edinburgh, UK, 1965.Google Scholar
Anshelevich, M., Appell polynomials and their relatives . Int. Math. Res. Not. IMRN 2004(2004), no. 65, 3469–3531.CrossRefGoogle Scholar
Anshelevich, M., Appell polynomials and their relatives. II: Boolean theory . Indiana Univ. Math. J. 58(2009a), no. 2, 929–968.CrossRefGoogle Scholar
Anshelevich, M., Appell polynomials and their relatives. III: conditionally free theory . Illinois J. Math. 53(2009b), no. 1, 39–66.CrossRefGoogle Scholar
Appell, P., Sur une classe de polynomes . Ann. Sci. Éc. Norm. Supér. (2) 2(1880), no. 9, 119–144.CrossRefGoogle Scholar
Arizmendi, O., Hasebe, T., Lehner, F., and Vargas, C., Relations between cumulants in noncommutative probability . Adv. Math. 282(2015), 56–92.CrossRefGoogle Scholar
Avitzour, D., Free products of C*-algebras . Trans. Amer. Math. Soc. 271(1982), no. 2, 423–435.Google Scholar
Bożejko, M., Positive definite functions on the free group and the noncommutative Riesz product . Boll. Unione Mat. Ital. A (6) 5(1986), no. 1, 13–21.Google Scholar
Bożejko, M., Leinert, M., and Speicher, R., Convolution and limit theorems for conditionally free random variables . Pacific J. Math. 175(1996), no. 2, 357–388.CrossRefGoogle Scholar
Ebrahimi-Fard, K. and Patras, F., Cumulants, free cumulants and half-shuffles . Proc. Roy. Soc. London Ser. A 471(2015), no. 2176, 20140843.Google ScholarPubMed
Ebrahimi-Fard, K. and Patras, F., A group-theoretical approach to conditionally free cumulants. Preprint, 2018a. arXiv:1806.06287 Google Scholar
Ebrahimi-Fard, K. and Patras, F., Monotone, free, and Boolean cumulants: a shuffle algebra approach . Adv. Math. 328(2018b), 112–132.CrossRefGoogle Scholar
Ebrahimi-Fard, K. and Patras, F., From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras. Preprint, 2019a. arXiv:1911.08766 Google Scholar
Ebrahimi-Fard, K. and Patras, F., Shuffle group laws. Applications in free probability . Proc. Lond. Math. Soc. 119(2019b), no. 3, 814–840.CrossRefGoogle Scholar
Ebrahimi-Fard, K., Patras, F., Tapia, N., and Zambotti, L., Hopf-algebraic deformations of products and Wick polynomials . Int. Math. Res. Not. IMRN 2020(2020), 10064–10099.CrossRefGoogle Scholar
Effros, E. G. and Popa, M., Feynman diagrams and Wick products associated with q-Fock space . Proc. Natl. Acad. Sci. USA 100(2003), no. 15, 8629–8633.CrossRefGoogle Scholar
Foissy, L., Bidendriform bialgebras, trees, and free quasi-symmetric functions . J. Pure Appl. Algebra 209(2007), no. 2, 439–459.CrossRefGoogle Scholar
Hairer, M. and Shen, H., A central limit theorem for the KPZ equation . Ann. Probab. 45(2017), no. 6B, 4167–4221.CrossRefGoogle Scholar
Hasebe, T. and Saigo, H., The monotone cumulants . Ann. Henri Poincaré B 47(2011), no. 4, 1160–1170.Google Scholar
Muraki, N., Monotonic independence, monotonic central limit theorem and monotonic law of small numbers . Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4(2001), no. 1, 39–58.CrossRefGoogle Scholar
Nica, A. and Speicher, R., Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
Peccati, G. and Taqqu, M. S., Wiener chaos: moments, cumulants and diagrams: a survey with computer implementation . Vol. 1, Springer Science & Business Media, Milan, 2011.Google Scholar
Reutenauer, C.. Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993.Google Scholar
Sarah, M. and Schurmann, M., Non-commutative stochastic independence and cumulants . Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20(2017), no. 2, 1750010.Google Scholar
Schützenberger, M. P., Sur une propriété combinatoire des algebres de Lie libres pouvant étre utilisée dans un probleme de mathématiques appliquées . Sém. Dubreil. Algèbre Théorie Nr. 12(1958), no. 1, 1–23.Google Scholar
Speicher, R., Free probability theory and non-crossing partitions . Sém. Lothar. Combin. 39(1997), 1–38.Google Scholar
Speicher, R. and Woroudi, R., Boolean convolution . In: Voiculescu, D. (ed.), Free probability theory. Papers from a workshop on random matrices and operator algebra free products, Toronto, Canada, Mars 1995, American Mathematical Society, Providence, RI, 1997, pp. 267–279.Google Scholar
Voiculescu, D.-V. (ed.), Free probability theory, Fields Institute Communications, 12, American Mathematical Society, Providence, RI, 1997.Google Scholar
Voiculescu, D.-V., Dykema, K., and Nica, A., Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992.Google Scholar
von Waldenfels, W., An approach to the theory of pressure broadening of spectral lines. In: Behara, M., Krickeberg, K., and Wolfowitz, J. (eds.), Probability and information theory II, Lecture Notes in Mathematics, 296, Springer, Berlin, 1973, pp. 19–69.CrossRefGoogle Scholar