Author:
Figueroa Héctor,Várilly Joseph C.,Gracia-Bondía José M.
Abstract
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.
Publisher
Universidad Nacional de Colombia
Reference26 articles.
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, Zurich, 2015.
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.
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.
4. W. E. Caswell and A. D. Kennedy, Simple approach to renormalization theory, Phys. Rev. D. 25 (1982), 392-408.
5. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, 1974.