Let $R$ be a commutative ring. A Newtonian coalgebra over $R$ is an $R$ -module $C$ which is simultaneously a coalgebra with comultiplication $\Delta\colon C\to C\otimes C$ and an algebra with multiplication $\cdot\colon C\otimes C\to C$ such that $\Delta$ is a derivation over $\cdot$ , that is, such that the identity $$ \Delta(u\cdot v) = \Delta(u)\cdot v + u\cdot\Delta(v) $$ 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 language which can explain iterated differentiation of trigonometric functions as well as Faà di Bruno's formula. 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 structure to the $\mathbf{cd}$ -index, a poset invariant generalizing the $f$ -vector of polytopes.

One example of a Newtonian coalgebra is the free associative algebra $R\langle\mathbf{a},\mathbf{b}\rangle$ of polynomials on the noncommuting variables $\mathbf{a}$ and $\mathbf{b}$ with coefficients in $R$ . The product is the ordinary noncommutative polynomial product, and the comultiplication is defined by setting $$ \Delta(u_1\cdots u_n) = \sum_{j\in[n]} u_1\cdots u_{i-1}\otimes u_{i+1}\cdots u_n $$ for each monomial and extending by linearity.

Bibliography

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 $\mathbf{cd}$ -index of polytopes. Discrete Comput. Geom., 27 (2002), no. 1, 3-28.

3

R. Ehrenborg and M. Readdy, Coproducts and the $\mathbf{cd}$ -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 bialgebras 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.