Servicios Personalizados
Revista
Articulo
Inglés (pdf)
Articulo en XML
Referencias del artículo
Traducción automática
Enviar articulo por emailIndicadores
-
Citado por SciELO -
Accesos
Links relacionados
-
Citado por Google -
Similares en
SciELO -
Similares en Google
Compartir
Permalink
Revista Colombiana de Matemáticas
versión impresa ISSN 0034-7426
Rev.colomb.mat. vol.56 no.1 Bogotá ene./jun. 2022 Epub 02-Ene-2024
https://doi.org/10.15446/recolma.v56n1.105611
ORIGINAL ARTICLES
Faà di Bruno Hopf algebras
Álgebras de Hopf de Faà di Bruno
1 Universidad de Costa Rica, San José, Costa Rica
2 Universidad de Zaragoza, Zaragoza, Spain
This is a short review on the Faá di Bruno formulas, implementing composition of real-analytic functions, and a Hopf algebra associated to such formulas. This structure provides, among several other things, a short proof of the Lie-Scheffers theorem, and relates the Lagrange inversion formulas with antipodes. It is also the maximal commutative Hopf subalgebra of the one used by Connes and Moscovici to study diffeomorphisms in a noncommutative geometry setting. The link of Faa di Bruno formulas with the theory of set partitions is developed in some detail.
Keywords: Faá di Bruno formula; Hopf algebras; partitions
Esta es una reseña corta sobre las fórmulas de Faá di Bruno, implementando composición de funciones analíticas reales, y algunas álgebras de Hopf asociadas a dichas fórmulas. Entre otras cosas, tal estructura permite una demostración corta del teorema de Lie y Scheffers, y establece la relación entre las fórmulas de inversión de Lagrange y los antípodas. Esta álgebra de Hopf es la subálgebra conmutativa maximal del álgebra introducida por Connes y Moscovici para estudiar difeomorfismos en el marco de la geometría no conmutativa. Asimismo, desarrollamos con cierto detalle el vínculo entre las fórmulas de Faà di Bruno y la teoría de particiones de conjuntos.
Palabras clave: Fórmula de Faà di Bruno; álgebras de Hopf; particiones
REFERENCES
1. Faà di Bruno Hopf algebras, Dyson-Schwinger equations, and Lie-Butcher Series, K. Ebrahimi-Fard and F. Fauvet, eds., IRMA Lectures in Mathematics and Theoretical Physics 21, EMS Press, Zürich, 2015. [ Links ]
2. J. C. Baez and J. Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, B. Engquist and W. Schmid, eds., Springer, Berlin (2001), 29-50. [ Links ]
3. J. F. Cariñena, K. Ebrahimi-Fard , H. Figueroa, and J. M. Gracia-Bondía, Hopf algebras in dynamical systems theory, Int. J. Geom. Methods Mod. Phys. 4 (2007), 577-646. [ Links ]
4. W. E. Caswell and A. D. Kennedy, Simple approach to renormalization theory, Phys. Rev. D. 25 (1982), 392-408. [ Links ]
5. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, 1974. [ Links ]
6. A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II: the (-function, diffeomorphisms and the renormalization group, Commun. Math. Phys. 216 (2001), 215-241. [ Links ]
7. A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys . 198 (1998), 198-246. [ Links ]
8. G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), 503-520. [ Links ]
9. A. D. D. Craik, Prehistory of Faà di Bruno's formula, Amer. Math. Monthly 112 (2005), 119-130. [ Links ]
10. J.-L. de Lagrange, Nouvelle méthode pour résoudre les équations littérales par la moyen des séries, Mém. Acad. Royale des Sciences et Belles-Lettres de Berlin 24 (1770), 251-326. [ Links ]
11. K. Ebrahimi-Fard and W. S. Gray, Center problem, Abel equation and the Faà di Bruno Hopf algebra for output feedback, Int. Math. Res. Notices 2017 (2017), 5415-5450. [ Links ]
12. K. Ebrahimi-Fard , A. Lundervold, and D. Manchon, Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras, Int. J. Alg. Comp. 24 (2014), 671-705. [ Links ]
13. H. Figueroa and J. M. Gracia-Bondía , Combinatorial Hopf algebras in quantum field theory, Rev. Math. Phys. 17 (2005), 881-976. [ Links ]
14. L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math. 218 (2008), 136-162. [ Links ]
15. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, MA, 1994. [ Links ]
16. M. Haiman and W. R. Schmitt, Incidence algebra antipodes and Lagrange inversion in one and several variables, J. Combin. Theory A. 50 (1989), 172-185. [ Links ]
17. E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976. [ Links ]
18. S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Contemp. Math. 6 (1982), 1-47. [ Links ]
19. J. Kock and M. Weber, Faà di Bruno for operads and internal algebras, J. London Math. Soc. 99 (2019), 919-944. [ Links ]
20. S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, 2nd edition, Birkhäuser, Boston, 2002. [ Links ]
21. D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, Math. Phys. Anal. Geom. 20 (2017), no. 16. [ Links ]
22. S. Lie and G. Scheffers, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893. [ Links ]
23. F. Menous, Formulas for the Connes-Moscovici Hopf algebra, C. R. Acad. Sci. (Paris) 341 (2005), 75-78. [ Links ]
24. A. Schreiber, Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions, Enumer. Combin. Appl. 1 (2021), 1. [ Links ]
25. R. P. Stanley, Generating functions in Studies in Combinatorics, Gian-Carlo Rota, ed., MAA, Washington, DC (1978), 100-141. [ Links ]
26. R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999. [ Links ]
Received: July 07, 2021; Accepted: January 27, 2022
This is an open-access article distributed under the terms of the Creative Commons Attribution License
