# 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 $\Delta\colon C\to C\otimes C$ 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 (http://planetmath.org/CdIndex), 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.

## References

• 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.
Title Newtonian coalgebra NewtonianCoalgebra 2013-03-22 16:48:36 2013-03-22 16:48:36 mps (409) mps (409) 6 mps (409) Definition msc 06A11 msc 16W30