Newtonian coalgebra

Let R be a commutative ring. A Newtonian coalgebra over R is an R-module C which is simultaneously a coalgebra with comultiplication Δ:CCC and an algebraMathworldPlanetmathPlanetmath with multiplication :CCC such that Δ is a derivationPlanetmathPlanetmath over , that is, such that the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


holds for any u and v in C. Newtonian coalgebras were introduced by Joni and Rota in [5], where they were called infinitesimal coalgebras. They reserved the term “Newtonian coalgebra” for the special case of the coalgebra of divided differences. This example was studied in more detail by Hirschhorn and Raphael [4]. Joni and Rota also showed that Newtonian coalgebras provide a languagePlanetmathPlanetmath which can explain iterated differentiation of trigonometric functions as well as Faà di Bruno’s formulaMathworldPlanetmathPlanetmath. See also the paper of Nichols and Sweedler [6] for more on trigonometric coalgebras.

A Newtonian coalgebra cannot have both a unit and a counit, so no Newtonian coalgebra is a Hopf algebra. However, Aguiar [1] developed a notion of antipode that makes sense for Newtonian coalgebras, leading to what he calls an infinitesimal Hopf algebra. Ehrenborg and Readdy [3] used Newtonian coalgebras to give an algebraic structurePlanetmathPlanetmath to the 𝐜𝐝-index (, a poset invariant generalizing the f-vector of polytopes.

One example of a Newtonian coalgebra is the free associative algebra R𝐚,𝐛 of polynomialsMathworldPlanetmathPlanetmath on the noncommuting variables 𝐚 and 𝐛 with coefficients in R. The productMathworldPlanetmathPlanetmath is the ordinary noncommutative polynomial product, and the comultiplication is defined by setting


for each monomial and extending by linearity.


  • 1 M. Aguiar, Infinitesimal Hopf algebras. New trends in Hopf algebra theory: (La Falda, 1999), 1–29, Contemp. Math., 267, Amer. Math. Soc., Providence, RI, 2000.
  • 2 M. Aguiar, Infinitesimal Hopf algebras and the 𝐜𝐝-index of polytopes. Discrete Comput. Geom., 27 (2002), no. 1, 3–28.
  • 3 R. Ehrenborg and M. Readdy, CoproductsMathworldPlanetmath and the 𝐜𝐝-index, J. Algebr. Comb., 8 (1998), 273–299.
  • 4 P. S. Hirschhorn and L. A. Raphael, Coalgebraic foundation of the method of divided differences, Adv. Math., 91 (1992), 75–135.
  • 5 S. A. Joni and G.-C. Rota, Coalgebras and bialgebrasPlanetmathPlanetmathPlanetmath in combinatorics, Stud. Appl. Math., 61 (1979), pp. 93–139.
  • 6 W. Nichols and M. Sweedler, Hopf algebras and combinatorics, in Proceedings of the conference on umbral calculus and Hopf algebras, ed. R. Morris, AMS, 1982.
Title Newtonian coalgebra
Canonical name NewtonianCoalgebra
Date of creation 2013-03-22 16:48:36
Last modified on 2013-03-22 16:48:36
Owner mps (409)
Last modified by mps (409)
Numerical id 6
Author mps (409)
Entry type Definition
Classification msc 06A11
Classification msc 16W30