Theory of Hopf algebras attached to group schemes

The Grothendieck Group of Hopf Algebras

Let H be a cosemisimple Hopf algebra over an algebraically closed field k. Assume that H contains a simple subcoalgebra of dimension 9. We show that if H has no simple subcoalgebras of even dimension then H contains either a grouplike element with order 2 or 3, a Hopf subalgebra of dimension 75, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. I...

The Formal Theory of Hopf Algebras

The category HopfC of Hopf monoids in a symmetric monoidal category C, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on C preserve directed colimits one has the following results: (1) If, in C, extremal epimorphisms are stable under tensor squaring, then HopfC is ...

Some Applications of Incidence Hopf Algebras to Formal Group Theory and Algebraic Topology

1. Introduction Hopf algebras are achieving prominence in combinatorics through the innuence of G.-C. Rota and his school, who developed the theory of incidence Hopf algebras (see 7], 15], 16]). The aim of this paper is to show that incidence Hopf algebras of partition lattices provide an eecient combinatorial framework for formal group theory and algebraic topology. We start by showing that th...

OB . K - Theory of Hopf Algebras

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...