Five interpretations of Faà di Bruno’s formula. (English) Zbl 1355.16031
Ebrahimi-Fard, Kurusch (ed.) et al., Faà di Bruno Hopf algebras, Dyson-Schwinger equations, and Lie-Butcher series. Based on a conference hosted by IRMA, Strasbourg, France, June 27–July 1, 2011. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-143-9/pbk; 978-3-03719-643-4/ebook). IRMA Lectures in Mathematics and Theoretical Physics 21, 91-147 (2015).
These lectures present the Faà di Bruno formula in five different settings; it also contains a short biography of Francesco Faà di Bruno and some historical remarks.
This formula gives the \(n\)-th derivative of the composite function \(f\circ g\) in terms of the derivatives of the smooth functions \(f\) and \(g\).
For the entire collection see [Zbl 1318.16001].
This formula gives the \(n\)-th derivative of the composite function \(f\circ g\) in terms of the derivatives of the smooth functions \(f\) and \(g\).
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- In terms of groups: this formula appears in the composition of the group \(G^{\mathrm{dif}}\) of formal diffeomorphisms of the line.
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- In terms of Hopf algebras: it appears in the coproduct of the Hopf algebra \(H_{\mathrm{FdB}}\) of coordinates of the pro-algebraic group \(G^{\mathrm{dif}}\). This related it to other Hopf algebras, bases on rooted trees or Feynman graphs. Moreover, \(H_{\mathrm{FdB}}\) admits a (still mysterious) noncommutative version.
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- In terms of Lie algebras: Faà di Bruno formula is related to a Lie subalgebra of the Witt (or of the Virasoro) Lie algebra. By Cartier-Quillen-Milnor-Moore’s theorem, this point of view is dual to the Hopf algebraic interpretation. This Lie algebra also admits extra structures, such as a prelie product or a brace bracket.
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- Combinatorially: \(H_{\mathrm{FdB}}\) is also the incidence algebra of a family of posets based on partitions.
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- In terms of operads: \(G^{\mathrm{dif}}\) is also a group associated to the operad Ass of associative algebras, and the Faà di Bruno formula reflects the composition of this operad.
For the entire collection see [Zbl 1318.16001].
Reviewer: Loïc Foissy (Calais)
MSC:
| 16T05 | Hopf algebras and their applications |
| 18D50 | Operads (MSC2010) |
| 16-03 | History of associative rings and algebras |
