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Faà di Bruno Hopf algebras
Álgebras de Hopf de Faà di Bruno
DOI:
https://doi.org/10.15446/recolma.v56n1.105611Palabras clave:
Faà di Bruno formula, Hopf algebras, partitions (en)Fórmula de Faà di Bruno, álgebras de Hopf, particiones (es)
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This is a short review on the Faà 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 Faà di Bruno formulas with the theory of set partitions is developed in some detail.
Esta es una reseña corta sobre las fórmulas de Faà 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 en cierto detalle el vínculo entre las fórmulas de Faà di Bruno y la teoría de particiones de conjuntos.
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