Split extension classifiers in the category of cocommutative 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...

Quasitriangular Structures on Cocommutative Hopf Algebras

The article is devoted to the describtion of quasitriangular structures (universal R-matrices) on cocommutative Hopf algebras. It is known that such structures are concentrated on finite dimensional Hopf subalgebras. In particular, quasitriangular structure on group algebra is defined by the pairs of normal inclusions of an finite abelian group and by invariant bimultiplicative form on it. The ...

Low Dimensional Cocommutative Connected Hopf Algebras

William M. Singer’s theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p3. The theory shows that there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of which is the Hopf algebra corre...

Cocommutative Calabi-yau Hopf Algebras and Deformations

The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal enveloping algebra of a finite dimensional Lie algebra g with a finite subgroup G of automorphisms of g is Calabi-Yau if and only if the universal en...

On split exact sequences Hopf algebras