 
Summary: INFINITESIMAL HOPF ALGEBRAS AND THE cdINDEX OF POLYTOPES.
MARCELO AGUIAR
Abstract. Innitesimal bialgebras were introduced by Joni and Rota [JR]. The basic theory of
these objects was developed in [A1, A2]. In this paper we present a simple proof of the existence of
the cdindex of polytopes, based on the theory of innitesimal Hopf algebras.
For the purpose of this work, the main examples of innitesimal Hopf algebras are provided by
the algebra P of all posets and the algebra kha; bi of noncommutative polynomials. We show that
kha; bi satises the following universal property: given a graded innitesimal bialgebra A and a
morphism of algebras A : A ! k, there exists a unique morphism of graded innitesimal bialgebras
: A ! kha; bi such that 1;0 = A , where 1;0 is evaluation at (1; 0). When the universal property
is applied to the algebra of posets and the usual zeta function P (P ) = 1, one obtains the abindex
of posets : P ! kha; bi.
The notion of antipode is used to dene an analog of the Mobius function of posets for more general
innitesimal Hopf algebras than P, and this in turn is used to dene a canonical innitesimal Hopf
subalgebra, called the eulerian subalgebra. All eulerian posets belong to the eulerian subalgebra of P.
The eulerian subalgebra of kha; bi is precisely the algebra spanned by c = a + b and d = ab + ba.
The existence of the cdindex of eulerian posets is then an immediate consequence of the simple fact
that eulerian subalgebras are preserved under morphisms of innitesimal Hopf algebras.
The theory also provides a version of the generalized DehnSommerville equations for more general
innitesimal Hopf algebras than kha; bi.
