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Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.56 no.1 Bogotá jan./jun. 2022 Epub 02-Jan-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
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Received: July 07, 2021; Accepted: January 27, 2022
This is an open-access article distributed under the terms of the Creative Commons Attribution License
