Faà di Bruno Hopf algebras

• Héctor Figueroa ^[1] ; Várilly, Joseph C. ^[1] ; Gracia-Bondía, José M. ^[2]
  1. [1] Universidad de Costa Rica

     Universidad de Costa Rica

     Hospital, Costa Rica

     
  2. [2] Universidad de Zaragoza

     Universidad de Zaragoza

     Zaragoza, España

     
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 56, Nº. 1, 2022, págs. 1-12
• Idioma: inglés
• DOI: 10.15446/recolma.v56n1.105611
• Títulos paralelos:
  • Álgebras de Hopf de Faà di Bruno
• Enlaces
  • Texto completo
• Resumen
  • español

    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.

  • English

    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.

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