Infinitesimal Hopf algebras and the cd-index of polytopes
A fundamental result in the combinatorial study of polytopes is that their flag \(f\)-vectors satisfy the generalized Dehn-Sommerville relations of \textit{M. M. Bayer} and \textit{L. J. Billera} [Invent. Math. 79, 143-157 (1985; Zbl 0543.52007)]. This is true more generally for the flag \(f\)-vectors of Eulerian posets -- the face lattice of a polytope is an Eulerian poset. The flag \(f\)-vector of a ranked poset is encoded algebraically by its \(\mathbf{ab}\)-index in the algebra \(k\langle{\mathbf a},{\mathbf b}\rangle\) of non-commutative polynomials in \({\mathbf a}\) and \({\mathbf b}\). In this setting, the generalized Dehn-Sommerville relations are equivalent to the flag \(f\)-vector being a member of the subalgebra \(k\langle{\mathbf c},{\mathbf d}\rangle\), where \({\mathbf c}={\mathbf a}+{\mathbf b}\) and \({\mathbf d}={\mathbf ab}+{\mathbf ba}\). The \(\mathbf{cd}\)-index of a polytope is its flag \(f\)-vector, expressed as an element in \(k\langle{\mathbf c},{\mathbf d}\rangle\). Ehrenborg deepened this algebraic study by showing that the association of a flag \(f\)-vector to a graded poset is a morphism of Hopf algebras from a Hopf algebra of posets to the Hopf algebra of quasi-symmetric functions [\textit{R. Ehrenborg}, Adv. Math. 119, 1-25 (1996; Zbl 0851.16033)]. Such Hopf morphisms underlying quasi-symmetric functions associated to combinatorial objects are ubiquitous in enumerative algebraic combinatorics and a general theory was developed to explain this by \textit{N. Bergeron} et al. [J. Comb. Theory, Ser. A 91, 84-110 (2000; Zbl 0969.05064)]. Under a different product, the Hopf algebra of quasi-symmetric functions is isomorphic to \(k\langle{\mathbf a},{\mathbf b}\rangle\) and its coproduct becomes a derivation, a structure that Aguiar studied, called an infinitesimal bialgebra. In this paper, Aguiar puts all these results on a solid algebraic framework, both explaining this ubiquity of quasi-symmetric functions in combinatorics and giving a new proof of the existence of the \(\mathbf{cd}\)-index. First, he defines an infinitesimal Hopf algebra (a graded infinitesimal bialgebra equipped with an antipode \(S\)) and then proves that \(k\langle{\mathbf a},{\mathbf b}\rangle\) possesses a universal property in the category of infinitesimal Hopf algebras: There is a canonical algebra map \(\zeta\colon k\langle{\mathbf a},{\mathbf b}\rangle\to k\), and given any algebra map \(\zeta_A\colon A\to k\), where \(A\) is an infinitesimal Hopf algebra, there is a unique morphism \(\psi\colon A\to k\langle{\mathbf a},{\mathbf b}\rangle\) such that \(\zeta_A=\zeta\psi\). Furthermore, given an algebra map \(\zeta_A\colon A\to k\), the map \(\mu:=\zeta S\), which maps \(A\to k\), provides \(A\) with an analog of the Möbius function. There is a unique largest subalgebra spanned by homogeneous elements \(a\) which satisfy \(\mu(x)+(-1)^n\zeta_A(x)=0\), where \(n\) is the degree of \(a\). This gives a canonical infinitesimal subalgebra of \(A\), called the Eulerian subalgebra. Furthermore, the map \(\psi\colon A\to k\langle{\mathbf a},{\mathbf b}\rangle\) restricts to a map of Eulerian subalgebras. This conceptual proof of the existence of the \(\mathbf{cd}\)-index is completed by Aguiar showing that the Eulerian subalgebra of \(k\langle{\mathbf a},{\mathbf b}\rangle\) is the algebra of \(\mathbf{cd}\)-polynomials, and that the Eulerian subalgebra of the Hopf algebra of posets contains all Eulerian posets.
scientific article; zbMATH DE number 1743174