| | |
Summary: INFINITESIMAL HOPF ALGEBRAS AND THE cd-INDEX OF POLYTOPES.
MARCELO AGUIAR
Abstract. In nitesimal bialgebras were introduced by Joni and Rota [J-R]. The basic theory of
these objects was developed in [A1, A2]. In this paper we present a simple proof of the existence of
the cd-index of polytopes, based on the theory of in nitesimal Hopf algebras.
For the purpose of this work, the main examples of in nitesimal Hopf algebras are provided by
the algebra P of all posets and the algebra kha; bi of noncommutative polynomials. We show that
kha; bi satis es the following universal property: given a graded in nitesimal bialgebra A and a
morphism of algebras A : A ! k, there exists a unique morphism of graded in nitesimal 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 ab-index
of posets : P ! kha; bi.
The notion of antipode is used to de ne an analog of the Mobius function of posets for more general
in nitesimal Hopf algebras than P, and this in turn is used to de ne a canonical in nitesimal 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 cd-index of eulerian posets is then an immediate consequence of the simple fact
that eulerian subalgebras are preserved under morphisms of in nitesimal Hopf algebras.
The theory also provides a version of the generalized Dehn-Sommerville equations for more general
in nitesimal Hopf algebras than kha; bi.
|
