Combinatorial Hopf Algebras of Simplicial Complexes

Combinatorial Hopf Algebras of Simplicial Complexes

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their f -vectors. We al...

Commutative combinatorial Hopf algebras

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its noncommutative dual is realized in three different ways, in particular, as the Grossman–Larson algebra of heap-ordered trees. ...

Homotopy Theory of Simplicial Abelian Hopf Algebras

We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Qu...

Combinatorial Hopf Algebras and Towers of Algebras

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras L n≥0 An can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of al...

On Isomorphism of Simplicial Complexes and Their Related Algebras