Infinitesimal Hopf Algebras and the cd-Index of Polytopes

Infinitesimal Hopf algebras

Infinitesimal bialgebras were introduced by Joni and Rota [J-R]. An infinitesimal bialgebra is at the same time an algebra and a coalgebra, in such a way that the comultiplication is a derivation. In this paper we define infinitesimal Hopf algebras, develop their basic theory and present several examples. It turns out that many properties of ordinary Hopf algebras possess an infinitesimal versi...

Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday’s realization of the associahedron, and J.-L. Loday and M. Ronco’s Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic k-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos’ brick polytopes. We describe combinatorially a natural surjecti...

Monotonicity of the Cd-index for Polytopes

We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional poly...

Convex Polytopes and the Index of Wiener–hopf Operators

We study the C∗-algebra of Wiener–Hopf operators AΩ on a cone Ω with polyhedral base P. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of AΩ, with liminary subquotients. One may define K-group valued “index maps” between the subquotients. These form the E1 term of the Atiyah–Hirzebruch type spectral sequence induced by the filtration. We sho...

On the Coderivations of Some Hopf Algebras