Infinitesimal Hopf Algebras and the cd-Index of Polytopes

Abstract

Infinitesimal bialgebras were introduced by Joni and Rota [JR]. The basic theory of these objects was developed in [Aff1] and [Aff2]. In this paper we present a simple proof of the existence of the <bo>cd</bo>-index of polytopes, based on the theory of infinitesimal Hopf algebras. For the purpose of this work, the main examples of infinitesimal Hopf algebras are provided by the algebra \ppp of all posets and the algebra k &lt;<bo>ab</bo>&gt; of noncommutative polynomials. We show that k &lt;<bo>ab</bo>&gt; satisfies the following universal property: given a graded infinitesimal bialgebra A and a morphism of algebras ζ A \colon A→ k , there exists a unique morphism of graded infinitesimal bialgebras ψ\colon A → k&lt;<bo>ab</bo>&gt; such that ζ_{1,0}ψ=ζ_A, where ζ_{1,0} is evaluation at (1,0). When the universal property is applied to the algebra of posets and the usual zeta function ζ_{\ppp}(P)=1, one obtains the \abindex of posets ψ\colon \ppp→k &lt;<bo>ab</bo>&gt;. The notion of antipode is used to define an analog of the Möbius function of posets for more general infinitesimal Hopf algebras than \ppp , and this in turn is used to define a canonical infinitesimal Hopf subalgebra, called the eulerian subalgebra. All eulerian posets belong to the eulerian subalgebra of \ppp . The eulerian subalgebra of k &lt;<bo>ab</bo>&gt; is precisely the algebra spanned by <bo>c=a+b</bo> and <bo>d=ab+ba</bo>. The existence of the <bo>cd</bo>-index of eulerian posets is then an immediate consequence of the simple fact that eulerian subalgebras are preserved under morphisms of infinitesimal Hopf algebras. The theory also provides a version of the generalized Dehn—Sommerville equations for more general infinitesimal Hopf algebras than k &lt;<bo>ab</bo>&gt;.