On the Classification of Strongly Graded Hopf Algebras

Some Algebraic Applications of Graded Categorical Group Theory

The homotopy classification of graded categorical groups and their homomorphisms is applied, in this paper, to obtain appropriate treatments for diverse crossed product constructions with operators which appear in several algebraic contexts. Precise classification theorems are therefore stated for equivariant extensions by groups either of monoids, or groups, or rings, or rings-groups or algebr...

On the Coderivations of Some Hopf Algebras

Cocommutative Hopf Algebras of Permutations and Trees

Consider the coradical filtration of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We show that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. We also sho...

Path Hopf Algebras and Co-path Hopf Algebras

The main goal is to study the Hopf algebra structure on quivers. The main result obtained by C. Cibils and M. Rosso is improved. That is, in the case of infinite dimensional isotypic components it is shown that the path coalgebra kQ admits a graded Hopf algebra structure if and only if Q is a Hopf quiver. All nonisomorphic point path Hopf algebras and point co-path Hopf algebras are found. The ...

Spectral Sequences for the Classification of Extensions of Hopf Algebras

We construct spectral sequences which provide a way to compute the cohomology theory that classifies extensions of graded connected Hopf algebras over a commutative ring as described by William M. Singer. Specifically, for (A,B) an abelian matched pair of graded connected R-Hopf algebras, we construct a pair of spectral sequences relating H∗(B,A) to Ext∗,∗ B (R,Cotor ∗,∗ A (R,R)). To illustrate...