Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 038, 17 pages      arXiv:2204.01445      https://doi.org/10.3842/SIGMA.2023.038
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

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

Kurusch Ebrahimi-Fard a, Frédéric Patras b, Nikolas Tapia c and Lorenzo Zambotti d
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 and Technische Universität Berlin, Berlin, Germany
d) LPSM, Sorbonne Université, CNRS, Université Paris Cité, 4 Place Jussieu, 75005 Paris, France

Received April 05, 2022, in final form May 29, 2023; Published online June 08, 2023

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.

Key words: non-commutative probability theory; non-commutative power series; moments and cumulants; combinatorial Hopf algebra; pre-Lie algebra.

pdf (425 kb)   tex (28 kb)  

References

  1. Anshelevich M., Appell polynomials and their relatives, Int. Math. Res. Not. 2004 (2004), 3469-3531, arXiv:math.CO/0311043.
  2. Anshelevich M., Appell polynomials and their relatives. II. Boolean theory, Indiana Univ. Math. J. 58 (2009), 929-968, arXiv:0712.4185.
  3. Anshelevich M., Appell polynomials and their relatives. III. Conditionally free theory, Illinois J. Math. 53 (2009), 39-66, arXiv:0803.4279.
  4. Anshelevich M., Free evolution on algebras with two states, J. Reine Angew. Math. 638 (2010), 75-101, arXiv:0803.4280.
  5. Arizmendi O., Hasebe T., Lehner F., Vargas C., Relations between cumulants in noncommutative probability, Adv. Math. 282 (2015), 56-92, arXiv:1408.2977.
  6. Baues H.J., The double bar and cobar constructions, Compositio Math. 43 (1981), 331-341.
  7. Brouder C., Runge-Kutta methods and renormalization, Eur. Phys. J. C 12 (2000), 521-534, arXiv:hep-th/9904014.
  8. Butcher J.C., B-series. Algebraic analysis of numerical methods, Springer Ser. Comput. Math., Vol. 55, Springer, Cham, 2021.
  9. Calaque D., Ebrahimi-Fard K., Manchon D., Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series, Adv. in Appl. Math. 47 (2011), 282-308, arXiv:0806.2238.
  10. Cartier P., Patras F., Classical Hopf algebras and their applications, Algebr. Appl., Vol. 29, Springer, Cham, 2021.
  11. Cayley A., A theorem on trees, Q. J. Math 23 (1889), 376-378.
  12. Chartier P., Hairer E., Vilmart G., Algebraic structures of B-series, Found. Comput. Math. 10 (2010), 407-427.
  13. Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203-242, arXiv:hep-th/9808042.
  14. Ebrahimi-Fard K., Patras F., Cumulants, free cumulants and half-shuffles, Proc. A. 471 (2015), 20140843, 18 pages, arXiv:1409.5664.
  15. Ebrahimi-Fard K., Patras F., The splitting process in free probability theory, Int. Math. Res. Not. 2016 (2016), 2647-2676, arXiv:1502.02748.
  16. Ebrahimi-Fard K., Patras F., Monotone, free, and boolean cumulants: a shuffle algebra approach, Adv. Math. 328 (2018), 112-132, arXiv:1701.06152.
  17. Ebrahimi-Fard K., Patras F., Shuffle group laws: applications in free probability, Proc. Lond. Math. Soc. 119 (2019), 814-840, arXiv:1704.04942.
  18. Ebrahimi-Fard K., Patras F., Quasi-shuffle algebras in non-commutative stochastic calculus, in Geometry and Invariance in Stochastic Dynamics, Springer Proc. Math. Stat., Vol. 378, Springer, Cham, 2021, 89-112, arXiv:2004.06945.
  19. Ebrahimi-Fard K., Patras F., Tapia N., Zambotti L., Hopf-algebraic deformations of products and Wick polynomials, Int. Math. Res. Not. 2020 (2020), 10064-10099, arXiv:1710.00735.
  20. Ebrahimi-Fard K., Patras F., Tapia N., Zambotti L., Wick polynomials in noncommutative probability: a group-theoretical approach, Canad. J. Math. 74 (2022), 1673-1699, arXiv:2001.03808.
  21. Hairer E., Lubich C., Wanner G., Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd ed., Springer Ser. Comput. Math., Vol. 31, Springer, Berlin, 2006.
  22. Hasebe T., Saigo H., Joint cumulants for natural independence, Electron. Commun. Probab. 16 (2011), 491-506, arXiv:1005.3900.
  23. Hasebe T., Saigo H., The monotone cumulants, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), 1160-1170, arXiv:0907.4896.
  24. Mingo J.A., Speicher R., Free probability and random matrices, Fields Inst. Monogr., Vol. 35, Springer, New York, 2017.
  25. Nica A., Speicher R., Lectures on the combinatorics of free probability, London Math. Soc. Lecture Note Ser., Vol. 335, Cambridge University Press, Cambridge, 2006.
  26. Voiculescu D., Free probability theory: random matrices and von Neumann algebras, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 227-241.

Previous article  Next article  Contents of Volume 19 (2023)